BRIGHAM EXPLORATION CO
SC 13D/A, 1999-04-08
CRUDE PETROLEUM & NATURAL GAS
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<PAGE>   1
                                -----------------

                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                -----------------

                                 SCHEDULE 13D/A
                    Under the Securities Exchange Act of 1934
                                (Amendment No. 2)

                           BRIGHAM EXPLORATION COMPANY
                                (Name of Issuer)

                                -----------------

                          COMMON STOCK, $.01 PAR VALUE
                         (Title of Class of Securities)

                                   109178 10 3
                                 (CUSIP Number)

                                  Julia Murray
                             General Counsel Finance
                      Enron Capital & Trade Resources Corp.
                                1400 Smith Street
                              Houston, Texas 77002
                                 (713) 853-6161
   (Name, Address and Telephone Number of Person Authorized to Receive Notices
                               and Communications)

                                 March 29, 1999
             (Date of Event which Requires Filing of this Statement)

If the filing person has previously filed a statement on Schedule 13G to report
the acquisition which is the subject of this Schedule 13D, and is filing this
schedule because of Sections 240,13d-1(e), 240.13d-(f) or 240.13 d-I (g), check
the following box: [ ]

NOTE: Schedules filed in paper format shall include a signed original and five
copies of the schedule, including all exhibits. See Section 240.13d-7(b) for
other parties to whom copies are to be sent.

*The remainder of this cover page shall be filled out for a reporting person's
initial filing on this form with respect to the subject class of securities, and
for any subsequent amendment containing information which would alter
disclosures provided in a prior cover page.

The information required on the remainder of this cover page shall not be deemed
"filed" for the purpose of Section 18 of the Securities Exchange Act of 1934
(the "Act") or otherwise subject to the liabilities of that section of the Act
but shall be subject to all other provisions of the Act (however, see the
Notes).

                               Page 1 of 12 pages

<PAGE>   2

CUSIP NO.: 109178 10 3

                                  SCHEDULE 13D

================================================================================
1        Name of Reporting Person; S.S. or IRS Identification Number

         Enron Capital & Trade Resources Corp.
- --------------------------------------------------------------------------------
2        Check the Appropriate Box If A Member of a Group* 
         (a) [ ] 
         (b) [X]
- --------------------------------------------------------------------------------
3        SEC Use Only

- --------------------------------------------------------------------------------
4        Source of Funds
         WC
- --------------------------------------------------------------------------------
5        Check Box If Disclosure of Legal Proceedings Is Required
         Pursuant to Item 2(d) or 2(e)

         [ ]
- --------------------------------------------------------------------------------
6        Citizenship or Place of Organization

         Delaware
- --------------------------------------------------------------------------------
7        Sole Voting Power
         0
- --------------------------------------------------------------------------------
8        Shared Voting Power

         2,052,632
- --------------------------------------------------------------------------------
9        Sole Dispositive Power
         0
- --------------------------------------------------------------------------------
10       Shared Dispositive Power

         2,052,632
- --------------------------------------------------------------------------------
11       Aggregate Amount Beneficially Owned by Each Reporting Person

         2,052,632
- --------------------------------------------------------------------------------
12       Check Box If the Aggregate Amount in Row (11) Excludes Certain Shares*

         [ ]
- --------------------------------------------------------------------------------
13       Percent of Class Represented by Amount in Row (11)

         14.3%
- --------------------------------------------------------------------------------
14       Type of Reporting Person

         CO
================================================================================

<PAGE>   3
CUSIP NO.:109178 10 3

                                  SCHEDULE 13D

================================================================================
1       Name of Reporting Person; S.S. or IRS Identification Number

        Enron Corp.
- --------------------------------------------------------------------------------
2       Check the Appropriate Box If A Member of a Group*
        (a) [ ]
        (b) [X]
- --------------------------------------------------------------------------------
3       SEC Use Only

- --------------------------------------------------------------------------------
4       Source of Funds

        WC
- --------------------------------------------------------------------------------
5       Check Box If Disclosure of Legal Proceedings Is Required
        Pursuant to Item 2(d) or 2(e)

        [ ]
- --------------------------------------------------------------------------------
6       Citizenship or Place of Organization

        Oregon
- --------------------------------------------------------------------------------
7       Sole Voting Power

        0
- --------------------------------------------------------------------------------
8.      Shared Voting Power

        2,052,632
- --------------------------------------------------------------------------------
9       Sole Dispositive Power

        0
- --------------------------------------------------------------------------------
10      Shared Dispositive Power

        2,052,632
- --------------------------------------------------------------------------------
11      Aggregate Amount Beneficially Owned by Each Reporting Person

        2,052,632
- --------------------------------------------------------------------------------
12      Check Box if the Aggregate Amount in Row (11) Excludes Certain Shares*

        [ ]
- --------------------------------------------------------------------------------
13      Percent of Class Represented by Amount in Row (11)

        14.3%
- --------------------------------------------------------------------------------
14      Type of Reporting Person

        CO
================================================================================

<PAGE>   4

CUSIP NO.:109178 10

                                  SCHEDULE 13D

================================================================================
1       Name of Reporting Person; S.S. or IRS Identification Number

        Joint Energy Development Investments II Limited Partnership
- --------------------------------------------------------------------------------
2       Check the Appropriate Box If Member of Group*
        (a)[ ]
        (b)[X]

- --------------------------------------------------------------------------------
3       SEC Use Only

- --------------------------------------------------------------------------------
4       Source of Funds
        WC 
- --------------------------------------------------------------------------------
5       Check Box If Disclosure of Legal Proceedings Is Required
        Pursuant to Item 2(d) or 2(e)

        [ ]
- --------------------------------------------------------------------------------
6       Citizenship or Place of Organization

        Delaware
- --------------------------------------------------------------------------------
7       Sole Voting Power

        0
- --------------------------------------------------------------------------------
8       Shared Voting Power

        1,539,474
- --------------------------------------------------------------------------------
9       Sole Dispositive Power

        0
- --------------------------------------------------------------------------------
10      Shared Dispositive Power

        1,539,474
- --------------------------------------------------------------------------------
11      Aggregate Amount Beneficially Owned by Each Reporting Person

        1,539,474
- --------------------------------------------------------------------------------
12      Check Box If the Aggregate Amount in Row (11) Excludes Certain Shares*

        [ ]
- --------------------------------------------------------------------------------
13      Percent of Class Represented by Amount in Row (11)

        11.0%
- --------------------------------------------------------------------------------
14      Type of Reporting Person

        PN
================================================================================


<PAGE>   5

                               AMENDMENT NO. 2 TO
                            STATEMENT ON SCHEDULE 13D

     INTRODUCTORY Note: All information herein with respect to Brigham
Exploration Company, Delaware corporation (the "Issuer"), and the common stock,
par value $.01 per share (the "Common Stock"), of the Issuer is to the best
knowledge and belief of the Reporting Entities, as defined herein. The joint
Schedule 13D of Enron Corp., an Oregon corporation ("Enron"), Enron Capital &
Trade Resources Corp., Delaware corporation ("ECT"), and Joint Energy
Development Investments II Limited Partnership, Delaware limited partnership
("JEDI II") filed September 2, 1998, as amended January 12, 1999, is further
amended as follows.

ITEM 2. IDENTITY AND BACKGROUND; ITEM 4. PURPOSE OF TRANSACTION AND ITEM 5.
INTEREST IN SECURITIES OF THE ISSUER.

     In connection with a loan modification effected as of March 26, 1999, the
two warrants to purchase an aggregate 1,000,000 shares of Common Stock of the
Issuer that were held by ECT Merchant Investments Corp., a Delaware corporation
("ECT Investments"), and JEDI II were amended and restated. The amended warrants
effect only two substantive changes. First, the exercise period was extended
from August 22, 2005 to August 22, 2008. Second, the exercise price was reduced
from $10.45 to $3.50 per share. The Registration Rights Agreement dated August
20, 1998 was amended to confirm that its provisions apply to the amended
warrants.

     Schedule I attached hereto sets forth certain additional information with
respect to each director and executive officer of ECT Investments, Enron, ECT
and Enron Capital II Corp. The filing of this statement on Schedule 13D shall
not be construed as an admission that Enron, ECT, or, to their knowledge, any
person listed in Schedule I is, for the purposes of Schedule 13(d) or 13(g) of
the Act, the beneficial owner of any securities covered by this Statement. None
of the Reporting Entities or, to its knowledge, any person listed on Schedule I
hereto has been, during the last five years (a) convicted of any criminal
proceeding (excluding traffic violations or similar misdemeanors) or (b) a party
to civil a proceeding of a judicial or administrative body of competent
jurisdiction and as a result of such proceeding was or is subject to a judgment,
decree or final order enjoining future violations of, or prohibiting or
mandating activities subject to, U.S. federal or state securities laws or
finding any violations with respect to such laws.

     On its Form 10-K for the year ended December 31, 1998, the Issuer reported
that 13,306,206 shares of Common Stock were outstanding as of March 26, 1999.
The following calculations of beneficial ownership assume that this number of
shares are currently outstanding. If JEDI II's amended warrant was fully
exercised, the 1,539,474 shares JEDI II would own in the aggregate would
represent approximately 11.0% of the 14,056,206 then-outstanding shares of
Common Stock. If both amended warrants were fully exercised, the 2,052,632
shares owned by JEDI II and ECT Investments would represent approximately 14.3%
of the 14,306,206 then-outstanding shares of Common Stock. By virtue of their
control relationships, ECT and Enron may be deemed to beneficially own the

                                                              Page 5 of 12 pages


<PAGE>   6
Common Stock and the warrants held by ECT Investments and JEDI II, but Enron and
ECT disclaim beneficial ownership of these securities. ECT Investments is not
listed as a Reporting Person because it would own less than 5% of the
then-outstanding shares of Common Stock if its amended warrant was fully
exercised.

     Other than the transaction described herein, none of the Reporting
Entities, nor to their knowledge any of the persons named in Schedule I hereto,
has effected any transaction in the Common Stock during the preceding sixty
days.

ITEM 7. MATERIAL TO BE FILED AS EXHIBITS.

     (i) Warrant No. A-3.

     (ii) Warrant No. A-4.

     (iii) Amendment to Registration Rights Agreement.


[signature page follows]




                                                              Page 6 of 12 pages

<PAGE>   7

                                   SIGNATURES

     After reasonable inquiry and to the best of the undersigned's knowledge and
belief, each of the undersigned hereby certifies that the information set forth
in this statement is true, complete and correct,

Dated: April 7, 1999.                 ENRON CAPITAL & TRADE RESOURCES CORP.

                                      By:
                                         ---------------------------------
                                              Peggy B. Menchaca
                                              Vice President and Secretary

Dated: April 7, 1999.                 ENRON CORP.

                                      By:
                                         ---------------------------------
                                              Peggy B. Menchaca
                                              Vice President and Secretary

Dated: April 7, 1999.                 JOINT ENERGY DEVELOPMENT INVESTMENTS II
                                      LIMITED PARTNERSHIP

                                      By: Enron Capital Management II Limited
                                          Partnership, its general partner

                                      By: Enron Capital II Corp., its general
                                          partner

                                      By:
                                         ---------------------------------
                                              Peggy B. Menchaca
                                              Vice President and Secretary




                                                              Page 7 of 12 pages
<PAGE>   8
                                   SCHEDULE I
                        DIRECTORS AND EXECUTIVE OFFICERS
                      ENRON CAPITAL & TRADE RESOURCES CORP.

<TABLE>
<CAPTION>
- --------------------------------------------------------------------------------
Name and Business Address         Citizenship      Position and Occupation
- --------------------------------------------------------------------------------
<S>                               <C>              <C>
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
Mark A. Frevert                     U.S.A.         Director; President-ECT
                                                   Europe and Managing Director
- --------------------------------------------------------------------------------
Mark E. Haedicke                    U.S.A.         Director; Managing Director
                                                   and General Counsel
- --------------------------------------------------------------------------------
Kevin P. Hannon                     U.S.A.         Director; President and Chief
                                                   Operating Officer
- --------------------------------------------------------------------------------
Kenneth D. Rice                     U.S.A.         Director; Chairman of the
                                                   Board; Chief Executive 
                                                   Officer and Managing 
                                                   Director; Chairman and Chief
                                                   Executive Officer-ECT North
                                                   America
- --------------------------------------------------------------------------------
J. Clifford Baxter                  U.S.A.         Vice Chairman
- --------------------------------------------------------------------------------
Jere C. Overdyke, Jr.               U.S.A.         President -- ECT Merchant
                                                   Finance
- --------------------------------------------------------------------------------
Philippe A. Bibi                    U.S.A.         Managing Director
- --------------------------------------------------------------------------------
W. Craig Childers                   U.S.A.         Managing Director
- --------------------------------------------------------------------------------
David W. Delainey                   U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Richard G. DiMichele                U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Jay L. Fitzgerald                   U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Michael J. Kopper                   U.S.A.         Managing Director
- --------------------------------------------------------------------------------
John J. Lavorato                    U.S.A          Managing Director
- --------------------------------------------------------------------------------
Danny J. McCarty                    U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Michael McConnell                   U.S.A.         Managing Director
- --------------------------------------------------------------------------------
J. Kevin McConville                 U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Gregory F. Piper                    U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Jeffrey A. Shankman                 U.S.A.         Managing Director
- --------------------------------------------------------------------------------
John R. Sherriff                    U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Colleen Sullivan-Shaklovitz         U.S.A.         Managing Director
- --------------------------------------------------------------------------------
Lawrence G. Whalley                 U.S.A.         Managing Director
- --------------------------------------------------------------------------------



</TABLE>


                                                              Page 8 of 12 pages

<PAGE>   9
                        DIRECTORS AND EXECUTIVE OFFICERS
                                   ENRON CORP.

<TABLE>
<CAPTION>
- --------------------------------------------------------------------------------
Name and Business Address         Citizenship      Position and Occupation
- --------------------------------------------------------------------------------
<S>                               <C>              <C>
Robert Belfer                                      Director; Chairman, President
767 Fifth Avenue, 46th Floor      U.S.A.           and Chief Executive Officer,
New York, New York 10153                           Belco Oil & Gas Corp.
- --------------------------------------------------------------------------------
Norman P. Blake, Jr.                               Director; Chairman, United
USF&G Corporation                 U.S.A.           States Fidelity and Guaranty
6225 Smith Avenue, LA0300                          Company
Baltimore, Maryland 21209
- --------------------------------------------------------------------------------
Ronnie C. Chan                                     Director; Chairman of Hang 
Hang Lung Development Company                      Lung Development Group
Limited                           U.S.A.
28/F, Standard Chartered Bank
Building
4 Des Vouex Road Central
Hong Kong
- --------------------------------------------------------------------------------
John H. Duncan
5851 San Felipe, Suite 850        U.S.A.           Director; Investments
Houston, Texas 77057
- --------------------------------------------------------------------------------
Joe H. Foy                                         Director; Retired Senior  
404 Highridge Drive               U.S.A.           Partner, Bracewell &      
Kerrville, Texas 78028                             Patterson, L.L.P.         
- --------------------------------------------------------------------------------
Wendy L. Gramm                                     Director; Former Chairman,  
PO Box 39134                      U.S.A.           U.S. Commodity Futures      
Washington, D.C. 20016                             Trading Commission          
- --------------------------------------------------------------------------------
Ken L. Harrison                                    Director; Vice Chairman of
121 S.W. Salmon Street            U.S.A.           Enron Corp. and Chairman
Portland, Oregon 97204                             and Chief Executive Officer,
                                                   Portland General Electric 
                                                   Company
- --------------------------------------------------------------------------------
Robert K. Jaedicke                                 Director, Professor         
Graduate School of Business       U.S.A.           (Emeritus), Graduate School 
Stanford University                                of Business, Stanford       
Stanford, California 94305                         University                  
- --------------------------------------------------------------------------------
</TABLE>



                                                              Page 9 of 12 pages

<PAGE>   10

<TABLE>
<CAPTION>
- --------------------------------------------------------------------------------
Name and Business Address         Citizenship      Position and Occupation
- --------------------------------------------------------------------------------
<S>                               <C>              <C>
Charles A. LeMaistre                               Director; President          
13104 Travis View Loop            U.S.A.           (Emeritus), University       
Austin, Texas 78732                                of Texas M.D. Anderson Cancer
                                                   Center                       
- --------------------------------------------------------------------------------
Jerome J. Meyer                                    Director; Chairman and Chief 
26600 S.W. Parkway                U.S.A.           Executive Officer, Tektronix,
Building 63, P.0. Box 1000                         Inc.                         
Wilsonville, Oregon 97070-1000                     
- --------------------------------------------------------------------------------
John A. Urquhart                                   Director; Senior Advisor to
John A. Urquhart Associates       U.S.A.           the Chairman of Enron Corp.;
111 Beach Road                                     President, John A. Urquhart 
Fairfield, Connecticut 06430                       Associates
- --------------------------------------------------------------------------------
John Wakeham                                       Director; Former U.K.
Pingleston House                                   Secretary of State for Energy
Old Alresford                     U.K.             and Leader of the Houses of
Hampshire S024 9TB                                 Commons and Lords
United Kingdom
- --------------------------------------------------------------------------------
Charls E. Walker                                   Director, Chairman, Walker & 
Walker & Walker, LLC              U.S.A.           Walker, LLC                  
10220 River Road, Suite 105                        
Potomac, Maryland 20854
- --------------------------------------------------------------------------------
Herbert S. Winokur, Jr.                            Director; President, Winokur
Winokur & Associates, Inc.        U.S.A.           & Associates, Inc.
30 East Elm Ct.                                                       
Greenwich, Connecticut 06830
- --------------------------------------------------------------------------------
Kenneth L. Lay                                     Director; Chairman and Chief
1400 Smith Street                 U.S.A.           Executive Officer, Enron 
Houston, Texas 77002                               Corp.
- --------------------------------------------------------------------------------
J. Clifford Baxter                                 Senior Vice President, 
1400 Smith Street                 U.S.A.           Corporate Development; Enron
Houston, Texas 77002                               Corp.
- --------------------------------------------------------------------------------
Richard B. Buy                    U.S.A.           Senior Vice President and 
1400 Smith Street                                  Chief Risk Officer
Houston, Texas 77002
- --------------------------------------------------------------------------------
</TABLE>



                                                             Page 10 of 12 pages
<PAGE>   11
<TABLE>
<CAPTION>
- --------------------------------------------------------------------------------
Name and Business Address         Citizenship      Position and Occupation
- --------------------------------------------------------------------------------
<S>                               <C>              <C>
Richard A. Causey                 U.S.A.           Senior Vice President, Chief
1400 Smith Street                                  Accounting, Information and
Houston, Texas 77002                               Administrative Officer,
                                                   Enron Corp.
- --------------------------------------------------------------------------------
James V. Derrick, Jr.                              Senior Vice President and 
1400 Smith Street                 U.S.A.           General Counsel, Enron Corp.
Houston, Texas 77002                                                  
- --------------------------------------------------------------------------------
Andrew S. Fastow                                   Senior Vice President and 
1400 Smith Street                 U.S.A.           Chief Financial Officer,  
Houston, Texas 77002                               Enron Corp.               
- --------------------------------------------------------------------------------
Mark A. Frevert                                    President -- ECT Europe and
1400 Smith Street                 U.S.A.           Managing Director, Enron
Houston, Texas 77002                               Capital & Trade Resources
                                                   Corp. and President and Chief
                                                   Executive Officer, Enron
                                                   Europe Limited
- --------------------------------------------------------------------------------
Stanley C. Horton                                  Chairman and Chief Executive
1400 Smith Street                 U.S.A.           Officer, Enron Gas Pipeline
Houston, Texas 77002                               Group
- --------------------------------------------------------------------------------
Rebecca P. Mark                                    Vice Chairman, Enron Corp.
1400 Smith Street                 U.S.A.           and Chairman and Chief  
Houston, Texas 77002                               Executive Officer, Azurix
                                                   Corp.
- --------------------------------------------------------------------------------
Lou L. Pai                                         Chairman and Chief Executive
1400 Smith Street                 U.S.A.           Officer, Enron Energy
Houston, Texas 77002                               Services, Inc.
- --------------------------------------------------------------------------------
Kenneth D. Rice                   U.S.A.           Chairman and Chief Executive
1400 Smith Street                                  Officer, Enron Capital & 
Houston, Texas 77002                               Trade Resources Corp. --  
                                                   North America
- --------------------------------------------------------------------------------
Jeffrey K. Skilling                                Director, President and Chief
1400 Smith Street                 U.S.A.           Operating Officer, Enron   
Houston, Texas 77002                               Corp.                      
- --------------------------------------------------------------------------------
Joseph W. Sutton                                   Chairman, President and 
1400 Smith Street                 U.S.A.           Chief Executive Officer and 
Houston, Texas 77002                               Chief Operating Officer, 
                                                   Enron International Inc. 
- --------------------------------------------------------------------------------
</TABLE>



                                                             Page 11 of 12 pages

<PAGE>   12
                        DIRECTORS AND EXECUTIVE OFFICERS
                             ENRON CAPITAL II CORP.

<TABLE>
<CAPTION>
- --------------------------------------------------------------------------------
Name and Business Address         Citizenship      Position and Occupation
- --------------------------------------------------------------------------------
<S>                               <C>              <C>
James V. Derrick, Jr.             U.S.A.           Director
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
Mark A. Frevert                   U.S.A.           Director
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
Kenneth D. Rice                   U.S.A.           Director; Chairman, Chief 
1400 Smith Street                                  Executive Officer and
Houston, Texas 77002                               Managing Director
- --------------------------------------------------------------------------------
J. Clifford Baxter                U.S.A.           President and Managing 
1400 Smith Street                                  Director
Houston, Texas 77002
- --------------------------------------------------------------------------------
Richard B. Buy                    U.S.A.           Managing Director
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
Andrew S. Fastow                  U.S.A.           Managing Director
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
Jeffrey McMahon                   U.S.A.           Managing Director, Finance 
1400 Smith Street                                  and Treasurer
Houston, Texas 77002
- --------------------------------------------------------------------------------
Mark E. Haedicke                  U.S.A.           Managing Director and General
1400 Smith Street                                  Counsel
Houston, Texas 77002
- --------------------------------------------------------------------------------
Jere C. Overdyke, Jr.             U.S.A.           Managing Director
1400 Smith Street
Houston, Texas 77002
- --------------------------------------------------------------------------------
</TABLE>



                                                             Page 12 of 12 pages

<PAGE>   13

                               INDEX TO EXHIBITS


         Exhibit                                          

(i)      Warrant No. A-3                                  
(ii)     Warrant No. A-4                                   
(iii)    Amendment to Registration Rights Agreement        





<PAGE>   1
                                                                     EXHIBIT (i)

                              WARRANT CERTIFICATE

Number of Warrants: 250,000                                  Warrant No. A-3

      This warrant certificate ("Warrant Certificate") certifies that, for
                                 value received,

                         ECT MERCHANT INVESTMENTS CORP.

is the registered holder of the number of warrants (the "Warrants") set forth
above. Each Warrant entitles the holder thereof, at any time or from time to
time during the Exercise Period, to purchase from the Company one fully paid and
nonassessable share of Common Stock at the Exercise Price, subject to adjustment
as provided herein. The Warrants constitute, as of the initial Issuance Date,
1.6402% of the outstanding Common Stock on fully diluted basis including, for
purposes of such calculation, the Acquired Shares and the Warrant Shares.
Initially capitalized terms used but not defined herein shall have the meanings
ascribed to them in the Securities Purchase Agreement. This Warrant Certificate
amends and restates Warrant No. A-1 issued to Enron Capital & Trade Resources
Corp. on August 20, 1998.

     "Common Stock" means the common stock, $.01 par value per share, of the
Company and such other class of securities as shall then represent the common
equity of the Company.

     "Company" means Brigham Exploration Company, Delaware corporation.

     "Exercise Period" means the period of time between the Funding Date, as
defined in the Securities Purchase Agreement and 5:00 p.m. (New York City time)
on the Expiration Date.

     "Exercise Price" subject in all circumstances to adjustment in accordance
with Section 2, means $3.50.

     "Expiration Date" means August 22, 2008.

     "Funding Date" is defined in the Securities Purchase Agreement.

     "IPO" shall mean the initial public offering of securities of the Company
consummated on May 24, 1997, pursuant to a registration statement filed under
the Securities Act of 1933, as amended, and the rules and regulations
promulgated thereunder.


<PAGE>   2
     "Issuance Date" means August 20, 1998.

     "Person" means any individual, corporation, company, partnership, joint
venture, trust, limited liability company, unincorporated organization or
government or any agency, instrumentality or political subdivision thereof, or
any other form of entity.

     "Price" means the average of the "high" and "low" prices as reported in The
Wall Street Journal's listing for such day (corrected for obvious typographical
errors) or if such shares are not reported in such listing, the average of the
reported "high" and "low" sales prices on the largest national securities
exchange (based on the aggregate dollar value of securities listed) on which
such shares are listed or traded, or if such shares are not listed or traded on
any national securities exchange, then the average of the reported "high" and
"low" sales prices for such shares in the over-the-counter market, as reported
on the National Association of Securities Dealers Automated Quotations System,
or, if such prices shall not be reported thereon, the average of the closing bid
and asked prices so reported, or, if such prices shall not be reported, then the
average of the closing bid and asked prices reported by the National Quotations
Bureau Incorporated, or, in all other cases, the Estimated Private Market Equity
Value divided by the number of outstanding shares (on a fully diluted basis
using the treasury stock method). The "average" Price per share for any period
shall be determined by dividing the sum of the Prices determined for the
individual trading days in such period by the number of trading days in such
period.

     "Securities Purchase Agreement" means the Securities Purchase Agreement,
dated as of August 20, 1998, among the Company, Enron Capital & Trade Resources
Corp. and Joint Energy Development Investments II Limited Partnership,
individually and as agent.

     1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in whole or in
part, at any time or from time to time, during the Exercise Period, by
presentation and surrender to the Company at its address set forth in Section 9
of (i) this Warrant Certificate with the Election To Exercise, attached hereto
as Exhibit A, duly completed and executed, and (ii) payment of the Exercise
Price, by bank draft or cashier's check, for the number of Warrants being
exercised. If the holder of this Warrant Certificate at any time exercises less
than all the Warrants, the Company shall issue to such holder a warrant
certificate identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number of Warrants previously exercised.
Likewise, upon the presentation and surrender of this Warrant Certificate to the
Company at its address set forth in Section 9 and at the request of the holder,
the Company will, without expense, at the option of the holder, issue to the
holder in substitution for this Warrant Certificate one or more warrant
certificates in identical form and for an aggregate number of Warrants equal to
the number of Warrants evidenced by this Warrant Certificate.





                                      -2-

<PAGE>   3
          (b) To the extent that the Warrants have not been exercised at or
prior to the Expiration Date, such Warrants shall expire and the rights of the
holder shall become void and of no effect.

     2. ANTIDILUTION ADJUSTMENTS. The shares of Common Stock purchasable on
exercise of the Warrants are shares of Common Stock as constituted as of the
Issuance Date. The number and kind of securities purchasable upon the exercise
of the Warrants, and the Exercise Price, shall be subject to adjustment from
time to time upon the happening of certain events, as follows:

         (a) Mergers, Consolidations and Reclassifications. In case of any
reclassification or change of outstanding securities issuable upon exercise of
the Warrants at any time after the Issuance Date (other than a change in par
value, or from par value to no par value, or from no par value to par value or
as result of a subdivision or combination to which subsection 2(b) applies), or
in case of any consolidation or merger of the Company with or into another
entity or other person (other than a merger with another entity or other person
in which the Company is the surviving corporation and which does not result in
any reclassification or change in the securities issuable upon exercise of this
Warrant Certificate), the holder of the Warrants shall have, and the Company, or
such successor corporation or other entity, shall covenant in the constituent
documents effecting any of the foregoing transactions that such holder does
have, the right to obtain upon the exercise of the Warrants, in lieu of each
share of Common Stock, other securities, money or other property theretofore
issuable upon exercise of a Warrant, the kind and amount of shares of stock,
other securities, money or other property receivable upon such reclassification,
change, consolidation or merger by a holder of the shares of Common Stock, other
securities, money or other property issuable upon exercise of a Warrant if the
Warrants had been exercised immediately prior to such reclassification, change,
consolidation or merger. The constituent documents effecting any such
reclassification, change, consolidation or merger shall provide for adjustments
which shall be as nearly equivalent as may be practicable to the adjustments
provided in this subsection 2(a). The provisions of this subsection 2(a) shall
similarly apply to successive reclassifications, changes, consolidations or
mergers.

          (b) Subdivisions and Combinations. If the Company, at any time after
the Issuance Date, shall subdivide its shares of Common Stock into a greater
number of shares, the Exercise Price in effect immediately prior to such
subdivision shall be proportionately reduced, and the number of shares of Common
Stock purchasable upon exercise of the Warrants shall be proportionately
increased, as at the effective date of such subdivision, or if the Company shall
take a record of holders of its Common Stock for such purpose, as at such record
date, whichever is earlier. If the Company, at any time after the Issuance Date,
shall combine its shares of Common Stock into a smaller number of shares, the
Exercise Price in effect immediately prior to such combination shall be
proportionately increased, and the number of shares of Common Stock purchasable
upon exercise of the Warrants shall be proportionately reduced, as at the
effective date



                                       -3-

<PAGE>   4
of such combination, or if the Company shall take a record of holders of its
Common Stock for purposes of such combination, as at such record date,
whichever is earlier.

          (c) Dividends and Distributions. If the Company at any time after the
Issuance Date shall declare a dividend on its Common Stock payable in stock or
other securities of the Company to the holders of its Common Stock, the holder
of this Warrant Certificate shall, without additional cost, be entitled to
receive upon any exercise of a Warrant, in addition to the Common Stock to which
such holder would otherwise be entitled upon such exercise, the number of shares
of stock or other securities which such holder would have been entitled to
receive if he had been a holder immediately prior to the record date for such
dividend (or, if no record date shall have been established, the payment date
for such dividend) of the number of shares of Common Stock purchasable on
exercise of such Warrant immediately prior to such record date or payment date,
as the case may be.

          (d) Certain Issuances of Securities. If the Company at any time after
the Issuance Date shall issue any additional shares of Common Stock (otherwise
than as provided in paragraphs (a) through (c) of this Section 2) at a price per
share less than the average Price per share of Common Stock for the 20 trading
days immediately preceding the date of the authorization of such issuance (the
"Market Price") by the Board of Directors, then the Exercise Price upon each
such issuance shall be adjusted to that price determined by multiplying the
Exercise Price by a fraction:

          i. the numerator of which shall be the sum of (1) the number of
     shares of Common Stock outstanding immediately prior to the issuance of
     such additional shares of Common Stock multiplied by the Market Price, and
     (2) the consideration, if any, received by the Company upon the issuance of
     such additional shares of Common Stock, and

          ii. the denominator of which shall be the Market Price multiplied by
     the total number of shares of Common Stock outstanding immediately after
     the issuance of such additional shares of Common Stock.

     No adjustments of the Exercise Price shall be made under this paragraph (d)
upon the issuance of any additional shares of Common Stock that (y) are issued
pursuant to thrift plans, stock purchase plans, stock bonus plans, stock option
plans, employee stock ownership plans and other incentive or profit sharing
arrangements for the benefit of employees ("Employee Benefit Plans") that
otherwise would cause an adjustment under this paragraph (d); provided that the
aggregate number of shares of Common Stock so issued (including the shares
issued pursuant to any options, rights or warrants or convertible or
exchangeable securities issued under such Employee Benefit Plans containing the
right to purchase shares of Common Stock) pursuant to Employee Benefit Plans
after the closing date of the IPO, as adjusted for any stock splits, stock
dividends or subdivisions or combinations of Common Stock prior to the
Expiration Date, shall not in the aggregate exceed 5%



                                       -4-

<PAGE>   5
of the Company's outstanding Common Stock at the time of such issuance; or (z)
are issued pursuant to any Common Stock Equivalent (as hereinafter defined) (i)
if upon the issuance of any such Common Stock Equivalent, any such adjustments
shall previously have been made pursuant to paragraph (e) of this Section 2 or
(ii) if no adjustment was required pursuant to paragraph (e) of this Section 2.

         (e) Common Stock Equivalents. If the Company shall, after the Issuance
Date, issue any security or evidence of indebtedness which is convertible into
or exchangeable for Common Stock ("Convertible Security"), or any warrant,
option or other right to subscribe for or purchase Common Stock or any
Convertible Security, other than pursuant to Employee Benefit Plans (together
with Convertible Securities, "Common Stock Equivalent"), or if, after any such
issuance, the price per share for which additional shares of Common Stock may be
issuable thereunder is amended, then the Exercise Price upon each such issuance
or amendment shall be adjusted as provided in subsection (d) on the basis that
(i) the maximum number of additional shares of Common Stock issuable pursuant to
all such Common Stock Equivalents shall be deemed to have been issued as of the
earlier of (a) the date on which the Company shall enter into a firm contract
for the issuance of such Common Stock Equivalent, or (b) the date of actual
issuance of such Common Stock Equivalent; and (ii) the aggregate consideration
for such maximum number of additional shares of Common Stock shall be deemed to
be the minimum consideration received and receivable by the Company for the
issuance of such additional shares of Common Stock pursuant to such Common Stock
Equivalent; provided, however, that no adjustment shall be made pursuant to this
subsection (e) unless the consideration received and receivable by the Company
per share of Common Stock for the issuance of such additional shares of Common
Stock pursuant to such Common Stock Equivalent is less than the Market Price. No
adjustment of the Exercise Price shall be made under this subsection (e) upon
the issuance of any Convertible Security which is issued pursuant to the
exercise of any warrants or other subscription or purchase rights therefor, if
any adjustment shall previously have been made in the Exercise Price then in
effect upon the issuance of such warrants or other rights pursuant to this
subsection (e).

          (f) Miscellaneous. The following provisions shall be applicable to the
making of adjustments in the Exercise Price hereinbefore provided in this
Section 2:

              i. The consideration received by the Company shall be deemed to be
     the following: (I) to the extent that any additional shares of Common Stock
     or any Common Stock Equivalent shall be issued for cash consideration, the
     consideration received by the Company therefor, or, if such additional
     shares of Common Stock or Common Stock Equivalent are offered by the
     Company for subscription, the subscription price, or, if such additional
     shares of Common Stock or Common Stock Equivalent are sold to underwriters
     or dealers for public offering without a subscription offering, the initial
     public offering price, in any such case excluding any amounts paid or
     receivable for accrued interest or accrued



                                      -5-

<PAGE>   6
     dividends and without deduction of any compensation, discounts, commissions
     or expenses paid or incurred by the Company for and in the underwriting of,
     or otherwise in connection with, the issue thereof, (II) to the extent that
     such issuance shall be for consideration other than cash, then, except as
     herein otherwise expressly provided, the fair value of such consideration
     at the time of such issuance as determined in good faith by the Board of
     Directors, as evidenced by certified resolution of the Board of Directors
     delivered to the holder of this Warrant Certificate setting forth such
     determination. The consideration for any additional shares of Common Stock
     issuable pursuant to any Common Stock Equivalent shall be the consideration
     received by the Company for issuing such Common Stock Equivalent, plus the
     additional consideration payable to the Company upon the exercise,
     conversion or exchange of such Common Stock Equivalent. In case of the
     issuance at any time of any additional shares of Common Stock or Common
     Stock Equivalent in payment or satisfaction of any dividend upon any class
     of stock other than Common Stock, the Company shall be deemed to have
     received for such additional shares of Common Stock or Common Stock
     Equivalent (which shall not be deemed to be dividend payable in, or other
     distribution of, Common Stock under subsection (c) above) consideration
     equal to the amount of such dividend so paid or satisfied.

              ii. Upon the expiration of the right to convert, exchange or
     exercise any Common Stock Equivalent the issuance of which effected an
     adjustment in the Exercise Price, if any such Common Stock Equivalent shall
     not have been converted, exercised or exchanged, the number of shares of
     Common Stock deemed to be issued and outstanding because they were issuable
     upon conversion, exchange or exercise of any such Common Stock Equivalent
     shall no longer be computed as set forth above, and the Exercise Price
     shall forthwith be readjusted and thereafter be the price which it would
     have been (but reflecting any other adjustments in the Exercise Price made
     pursuant to the provisions of subsection (d) after the issuance of such
     Common Stock Equivalent) had the adjustment of the Exercise Price made upon
     the issuance or sale of such Common Stock Equivalent been made on the basis
     of the issuance only of the number of additional shares of Common Stock
     actually issued upon exercise, conversion or exchange of such Common Stock
     Equivalent and thereupon only the number of additional shares of Common
     Stock actually so issued shall be deemed to have been issued and only the
     consideration actually received by the Company (computed as in subparagraph
     (i) of this paragraph (f)) shall be deemed to have been received by the
     Company.

              iii. The number of shares of Common Stock at any time outstanding
     shall not include any shares thereof then directly or indirectly owned or
     held by or for the account of the Company or its wholly owned subsidiaries.



                                       -6-

<PAGE>   7
              iv. For the purposes of this Section 2, the term "shares of Common
     Stock" shall mean shares of (i) the class of stock designated as the Common
     Stock at the date hereof or (ii) any other class of stock resulting from
     successive changes or reclassifications of such shares consisting solely of
     changes in par value, or from par value to no par value, or from no par
     value to par value. If at any time, because of an adjustment pursuant to
     subsection (a), the Warrants shall entitle the holders to purchase any
     securities other than shares of Common Stock, thereafter the number of such
     other securities so purchasable upon exercise of each Warrant and the
     Exercise Price of such securities shall be subject to adjustment from time
     to time in a manner and on terms as nearly equivalent as practicable to the
     provisions with respect to the Common Stock contained in this Section 2.

          (g) Calculation of Exercise Price. The Exercise Price in effect from
time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

     3. NOTICE OF ADJUSTMENTS.  Whenever the Exercise Price or the number of
shares of Common Stock is required to be adjusted as provided in Section 2, the
Company shall forthwith compute the adjusted Exercise Price or the number of
shares of Common Stock issuable and shall prepare and mail to the holder hereof
a certificate setting forth such adjusted Exercise Price or such number of
shares of Common Stock, showing in reasonable detail the facts upon which the
adjustment is based.

     4. VOLUNTARY REDUCTION. (a) The Company may at its option, but shall not be
obligated to, at any time during the term of the Warrants, reduce the then
current Exercise Price by any amount selected by the Board of Directors;
provided that if the Company elects so to reduce the then current Exercise
Price, such reduction shall be irrevocable during its effective period and
remain in effect for minimum of 30 days following the date of such election,
after which time the Company may, at its option, reinstate the Exercise Price in
effect prior to such reduction. Whenever the Exercise Price is reduced, the
Company shall mail to the holder notice of the reduction at least 30 days before
the date the reduced Exercise Price takes effect, stating the reduced Exercise
Price and the period for which such reduced Exercise Price will be in effect.

         (b) The Company may make such decreases in the Exercise Price, in
addition to those required or allowed by this Section 4, as shall be determined
by it, as evidenced by a certified resolution of the Board of Directors
delivered to the holders, to be advisable to avoid or diminish any income tax to
the holder resulting from any dividend or distribution of stock or issuance of
rights or warrants to purchase or subscribe for stock or from any event treated
as such for income tax purposes.



                                      -7-
<PAGE>   8
     5. NOTICES TO WARRANT HOLDERS. In the event:

          (a) the Company shall authorize any consolidation or merger to which
the Company is a party and for which approval of any stockholders of the Company
is required, or of the conveyance or sale of all or substantially all of the
assets of the Company, or of any reclassification or change of the Common Stock
or other securities issuable upon exercise of the Warrants (other than a change
in par value, or from par value to no par value, or from no par value to par
value or as a result of subdivision or combination), or a tender offer or
exchange offer for shares of Common Stock (or other securities issuable upon the
exercise of the Warrants); or

          (b) the Company shall declare any dividend (or any other distribution)
on the Common Stock or any other class of its capital stock; or

          (c) the Company shall authorize the granting to the holders of Common
Stock or any other class of its capital stock of rights or warrants to subscribe
for or purchase any shares of any class or series of capital stock or any other
securities convertible into or exchangeable for shares of stock; or

          (d) of the voluntary or involuntary dissolution, liquidation or
winding up of the Company;

     then the Company shall cause to be sent to the holder hereof, at least 30
days prior to the applicable record date hereinafter specified, or promptly in
the case of events for which there is no record date, a written notice stating
(x) the date for the determination of the holders of record of shares of Common
Stock (or other securities issuable upon the exercise of the Warrants) entitled
to receive any such dividends or other distribution, (y) the initial expiration
date set forth in any tender offer or exchange offer for shares of Common Stock
(or other securities issuable upon the exercise of the Warrants), or (z) the
date on which any of the events specified in subsections (a)-(d) is expected to
become effective or consummated, and the date as of which it is expected that
holders of record of shares of Common Stock (or other securities issuable upon
the exercise of the Warrants) shall be entitled to exchange such shares for
securities or other property, if any, deliverable upon any such event. Failure
to give such notice or any defect therein shall not affect the legality or
validity of any such event, or the vote upon any such action.

     6. REPORTS TO WARRANT HOLDERS. The Company will cause to be delivered, by
first-class mail, postage prepaid, to the holder at such holder's address
appearing hereon, or such other address as the holder shall specify, a copy of 
any reports delivered by the Company to the holders of Common Stock.



                                       -8-

<PAGE>   9
     7. COVENANTS OF THE COMPANY. The Company covenants and agrees that:

          (a) Until the Expiration Date, the Company shall at all times reserve
and keep available, free from preemptive rights, out of the aggregate of its
authorized but unissued Common Stock (and other securities), for the purpose of
enabling it to satisfy any obligation to issue shares of Common Stock (and other
securities) upon the exercise of the Warrants, the number of shares of Common
Stock (and other securities) issuable upon the exercise of such Warrants.

          (b) The Company shall pay all expenses, taxes and other charges
payable in connection with the preparation, issuance and delivery of new warrant
certificates on transfer of the Warrants.

          (c) All Common Stock (and other securities) which may be issued upon
exercise of the Warrants shall upon issuance be validly issued, fully paid,
non-assessable and free from all preemptive rights and all taxes, liens and
charges with respect to the issuance thereof, and will not be subject to any
restrictions on voting or transfer thereof except as set forth in any
stockholders agreement.

          (d) All original issue taxes payable in respect of the issuance of
shares of Common Stock to the registered holder hereof upon the exercise of the
Warrants shall be borne by the Company; provided, that the Company shall not be
required to pay any tax or charge imposed in connection with any transfer
involved in the issuance of any certificate representing shares of Common Stock
(and other securities) in any name other than that of the registered holder
hereof, and in such case the Company shall not be required to issue or deliver
any certificate representing shares of Common Stock (and other securities) until
such tax or other charge has been paid or it has been established to the
Company's satisfaction that no such tax or charge is due.

          (e) As soon as practicable after the receipt from the holder of this
Warrant Certificate of notice of the exercise of a number of warrants sufficient
to require filing under the Hart-Scott-Rodino Antitrust Improvements Act of 1976
and the rules, regulations and formal interpretations thereunder, as amended
from time to time (the "HSR Act"), but in any event no later than the 10th
business day after receipt of such notice, the Company will (i) prepare and
file with the Antitrust Division of the Department of Justice (the "DOJ") and
the Federal Trade Commission (the "FTC") the Notification and Report Form
(accompanied by all documentary attachments contemplated thereby) required by
the HSR Act, (ii) upon the request of the holder, request early termination of
the waiting period imposed by the HSR Act, (iii) coordinate and cooperate with
the holder in responding to formal and informal requests for additional
information and documentary material from the DOJ and the FTC in connection with
such filing, (iv) use its best efforts to take, or cause to be taken, all
reasonable action and to do, or cause to be done, all things necessary and
appropriate to permit the issuance to the holder of the shares of Common Stock
issuable upon the



                                      -9-

<PAGE>   10
exercise of the warrants with respect to which any filing is required under the
HSR Act, and (v) reimburse the holder for the entire amount of any filing fee or
any other costs and expenses incurred by the holder in connection therewith
(including legal fees), or as required to be paid under the HSR Act.

          (f) QUOTATION ON NASDAQ.  The Company shall maintain the designation
and quotations, or listing, of its Common Stock on the NASDAQ national market
(or on the New York Stock Exchange or the American Stock Exchange) until the
date on which none of the Warrants or Warrant Shares remain outstanding.

     8. NO RIGHTS AS STOCKHOLDER.  The holder of the Warrants shall not, by
virtue of holding such Warrants, be entitled to any rights of a stockholder of
the Company either at law or in equity, and the rights of the holder of the
Warrants are limited to those expressed herein.

     9. NOTICES. All notices provided for hereunder shall be in writing and may
be given by registered or certified mail, return receipt requested, telex,
telegram, telecopier, air courier guaranteeing overnight delivery of personal
delivery, if to the holder at the following address:

          ECT Merchant Investments Corp.
          1400 Smith Street
          Houston, Texas 77002
          Attention: Donna Lowry
          Telecopier: (713) 646-4039

     and, if to the Company:

          Brigham Exploration Company
          6300 Bride Point Parkway
          Building 2, Suite 500
          Austin, Texas 78730
          Attention: Craig M. Fleming
          Telecopier: (512) 472-3400

     10. GOVERNING LAW. This Warrant Certificate shall be governed by and
construed in accordance with the laws of the State of Texas without regard to
principles of conflict of laws.

     11. LOST, STOLEN, MUTILATED OR DESTROYED WARRANT CERTIFICATES. Upon receipt
by the Company of evidence reasonably satisfactory to it of the ownership of and
the loss, theft, destruction


                                      -10-

<PAGE>   11
or mutilation of any Warrant Certificate, then, in the absence of notice to the
Company that such Warrant Certificate has been acquired by a bona fide
purchaser, the Company shall execute and deliver, in exchange for or in lieu of
the lost, stolen, destroyed or mutilated Warrant Certificate, a substitute
Warrant Certificate of the same tenor and evidencing a like number of Warrants.

          12. ASSIGNMENT. The holder of this Warrant Certificate shall be
entitled, without obtaining the consent of the Company, to transfer or assign
its rights, title and interest in (and rights, title and interest under) this
Warrant Certificate in whole or in part to any Person or Persons. Upon surrender
of this Warrant Certificate to the Company, with the Transfer Form annexed
hereto as Exhibit B duly executed, the Company shall, without charge, execute
and deliver a new warrant certificate or warrant certificates, identical in form
to this Warrant Certificate, evidencing the number of Warrants being transferred
pursuant to the Transfer Form in the name of the assignee or assignees named in
such Transfer Form. If the holder's entire interest is not being assigned, the
Company shall, without charge, execute and deliver one or more new warrant
certificates identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number being transferred pursuant to the
Transfer Form, and this Warrant Certificate shall promptly be canceled. The
terms and provisions of this Warrant Certificate shall inure to the benefit of
the holder and its successors and assigns and shall be binding upon the Company
and its successors and assigns, including, without limitation, any Person
succeeding the Company by merger, consolidation or acquisition of all or
substantially all of the Company's assets.



                                      -11-
<PAGE>   12

          IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to
be executed as of March 26, 1999, by the undersigned, thereunto duly authorized.

                                        BRIGHAM EXPLORATION COMPANY

                                        By:
                                           ------------------------------
                                             Karen E. Lynch
                                             Vice President


                                      -12-

<PAGE>   13

                                    EXHIBIT A

                              ELECTION TO EXERCISE
          [To be executed on exercise of the Warrants evidenced by this
                              Warrant Certificate]

TO:       Brigham Exploration Company

          The undersigned, the holder of the Warrants evidenced by the attached
Warrant Certificate, hereby irrevocably elects to exercise Warrants, and
herewith makes payment of ___________________ ($_________) representing the 
aggregate Exercise Price thereof, and requests that the certificate representing
the securities issuable hereunder be issued in the name of ____________________
and delivered to ________________, whose address is_____________________________

          Dated:
                -------------------

                      Name of Registered Holder:
                                                -------------------------------
                      Signature:
                                -----------------------------------------------
                      Title:
                            ---------------------------------------------------
                      Address:
                              -------------------------------------------------

NOTICE:- The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.


                                      -13-

<PAGE>   14
                                   EXHIBIT B

                                  TRANSFER FORM
      [To be executed only upon transfer of the Warrants evidenced by this
                              Warrant Certificate]

          FOR VALUE RECEIVED, the undersigned hereby sells, assigns and
transfers unto _____________________________the Warrants represented by the 
within Warrant Certificate, together with all right, title and interest therein,
and does hereby irrevocably constitute and appoint ____________________________
Attorney-in-Fact, to transfer same on the books of the Company with full power
of substitution in the premises.

          Dated:
                -------------------

                      Name of Registered Holder:
                                                -------------------------------
                      Signature:
                                -----------------------------------------------
                      Title:
                            ---------------------------------------------------
                      Address:
                              -------------------------------------------------

WITNESS:


- -----------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.


                                      -14-


<PAGE>   1
                                                                    EXHIBIT (ii)

                              WARRANT CERTIFICATE

Number of Warrants: 750,000                                     Warrant No. A-4

      This warrant certificate ("Warrant Certificate") certifies that, for
                                 value received,

          JOINT ENERGY DEVELOPMENT INVESTMENTS II LIMITED PARTNERSHIP

is the registered holder of the number of warrants (the "Warrants") set forth
above. Each Warrant entitles the holder thereof, at any time or from time to
time during the Exercise Period, to purchase from the Company one fully paid and
nonassessable share of Common Stock at the Exercise Price, subject to adjustment
as provided herein. The Warrants constitute, as of the initial Issuance Date,
4.9206% of the outstanding Common Stock on fully diluted basis including, for
purposes of such calculation, the Acquired Shares and the Warrant Shares.
Initially capitalized terms used but not defined herein shall have the meanings
ascribed to them in the Securities Purchase Agreement. This Warrant Certificate
amends and restates Warrant No. A-2 issued to Joint Energy Development
Investments II Limited Partnership on August 20, 1998.

          "Common Stock" means the common stock, $.01 par value per share, of
the Company and such other class of securities as shall then represent the
common equity of the Company.

          "Company" means Brigham Exploration Company, Delaware corporation.

          "Exercise Period" means the period of time between the Funding Date,
as defined in the Securities Purchase Agreement and 5:00 p.m. (New York City
time) on the Expiration Date.

          "Exercise Price" subject in all circumstances to adjustment in
accordance with Section 2, means $3.50.

          "Expiration Date" means August 22, 2008.

          "Funding Date" is defined in the Securities Purchase Agreement.

          "IPO" shall mean the initial public offering of securities of the
Company consummated on May 24, 1997, pursuant to registration statement filed
under the Securities Act of 1933, as amended, and the rules and regulations
promulgated thereunder.


<PAGE>   2

          "Issuance Date" means August 20, 1998.

          "Person" means any individual, corporation, company, partnership,
joint venture, trust limited liability company, unincorporated organization or
government or any agency, instrumentality or political subdivision thereof, or
any other form of entity.

          "Price" means the average of the "high" and "low" prices as reported
in The Wall Street Journal's listing for such day (corrected for obvious
typographical errors) or if such shares are not reported in such listing, the
average of the reported "high" and "low" sales prices on the largest national
securities exchange (based on the aggregate dollar value of securities listed)
on which such shares are listed or traded, or if such shares are not listed or
traded on any national securities exchange, then the average of the reported
"high" and "low" sales prices for such shares in the over the-counter market, as
reported on the National Association of Securities Dealers Automated Quotations
System, or, if such prices shall not be reported thereon, the average of the
closing bid and asked prices so reported, or, if such prices shall not be
reported, then the average of the closing bid and asked prices reported by the
National Quotations Bureau Incorporated, or, in all other cases, the Estimated
Private Market Equity Value divided by the number of outstanding shares (on
a fully diluted basis using the treasury stock method). The "average" Price per
share for any period shall be determined by dividing the sum of the Prices
determined for the individual trading days in such period by the number of
trading days in such period.

           "Securities Purchase Agreement" means the Securities Purchase
Agreement, dated as of August 20, 1998, among the Company, Enron Capital & Trade
Resources Corp. and Joint Energy Development Investments II Limited
Partnership, individually and as agent.

          1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in whole or
in part, at any time or from time to time, during the Exercise Period, by
presentation and surrender to the Company at its address set forth in Section 9
of (i) this Warrant Certificate with the Election To Exercise, attached hereto
as Exhibit A, duly completed and executed, and (ii) payment of the Exercise
Price, by bank draft or cashier's check, for the number of Warrants being
exercised. If the holder of this Warrant Certificate at any time exercises less
than all the Warrants, the Company shall issue to such holder warrant
certificate identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number of Warrants previously exercised.
Likewise, upon the presentation and surrender of this Warrant Certificate to the
Company at its address set forth in Section 9 and at the request of the holder,
the Company will, without expense, at the option of the holder, issue to the
holder in substitution for this Warrant Certificate one or more warrant
certificates in identical form and for an aggregate number of Warrants equal to
the number of Warrants evidenced by this Warrant Certificate.



                                       -2-
<PAGE>   3

          (b) To the extent that the Warrants have not been exercised at or
prior to the Expiration Date, such Warrants shall expire and the rights of the
holder shall become void and of no effect.

     2. ANTIDILUTION ADJUSTMENTS. The shares of Common Stock purchasable on
exercise of the Warrants are shares of Common Stock as constituted as of the
Issuance Date. The number and kind of securities purchasable upon the exercise
of the Warrants, and the Exercise Price, shall be subject to adjustment from
time to time upon the happening of certain events, as follows:

          (a) Mergers. Consolidations and Reclassifications. In case of any
reclassification or change of outstanding securities issuable upon exercise of
the Warrants at any time after the Issuance Date (other than a change in par
value, or from par value to no par value, or from no par value to par value or
as result of subdivision or combination to which subsection 2(b) applies), or in
case of any consolidation or merger of the Company with or into another entity
or other person (other than a merger with another entity or other person in
which the Company is the surviving corporation and which does not result in any
reclassification or change in the securities issuable upon exercise of this
Warrant Certificate), the holder of the Warrants shall have, and the Company, or
such successor corporation or other entity, shall covenant in the constituent
documents effecting any of the foregoing transactions that such holder does
have, the right to obtain upon the exercise of the Warrants, in lieu of each
share of Common Stock, other securities, money or other property theretofore
issuable upon exercise of a Warrant, the kind and amount of shares of stock,
other securities, money or other property receivable upon such reclassification,
change, consolidation or merger by a holder of the shares of Common Stock, other
securities, money or other property issuable upon exercise of a Warrant if the
Warrants had been exercised immediately prior to such reclassification, change,
consolidation or merger. The constituent documents effecting any such
reclassification, change, consolidation or merger shall provide for adjustments
which shall be as nearly equivalent as may be practicable to the adjustments
provided in this subsection 2(a). The provisions of this subsection 2(a) shall
similarly apply to successive reclassifications, changes, consolidations or
mergers.

          (b) Subdivisions and Combinations. If the Company, at any time after
the Issuance Date, shall subdivide its shares of Common Stock into a greater
number of shares, the Exercise Price in effect immediately prior to such
subdivision shall be proportionately reduced, and the number of shares of Common
Stock purchasable upon exercise of the Warrants shall be proportionately
increased, as at the effective date of such subdivision, or if the Company shall
take a record of holders of its Common Stock for such purpose, as at such record
date, whichever is earlier. If the Company, at any time after the Issuance Date,
shall combine its shares of Common Stock into a smaller number of shares, the
Exercise Price in effect immediately prior to such combination shall be
proportionately increased, and the number of shares of Common Stock purchasable
upon exercise of the Warrants shall be proportionately reduced, as at the
effective date



                                       -3-

<PAGE>   4

of such combination, or if the Company shall take a record of holders of its
Common Stock for purposes of such combination, as at such record date, whichever
is earlier.

          (c) Dividends and Distributions. If the Company at any time after the
Issuance Date shall declare a dividend on its Common Stock payable in stock or
other securities of the Company to the holders of its Common Stock, the holder
of this Warrant Certificate shall, without additional cost, be entitled to
receive upon any exercise of a Warrant, in addition to the Common Stock to which
such holder would otherwise be entitled upon such exercise, the number of shares
of stock or other securities which such holder would have been entitled to
receive if he had been a holder immediately prior to the record date for such
dividend (or, if no record date shall have been established, the payment date
for such dividend) of the number of shares of Common Stock purchasable on
exercise of such Warrant immediately prior to such record date or payment date,
as the case may be.

          (d) Certain Issuances of Securities. If the Company at any time after
the Issuance Date shall issue any additional shares of Common Stock (otherwise
than as provided in paragraphs (a) through (c) of this Section 2) at a price per
share less than the average Price per share of Common Stock for the 20 trading
days immediately preceding the date of the authorization of such issuance (the
"Market Price") by the Board of Directors, then the Exercise Price upon each
such issuance shall be adjusted to that price determined by multiplying the
Exercise Price by fraction:

              i. the numerator of which shall be the sum of (1) the number of
     shares of Common Stock outstanding immediately prior to the issuance of
     such additional shares of Common Stock multiplied by the Market Price, and
     (2) the consideration, if any, received by the Company upon the issuance of
     such additional shares of Common Stock, and

              ii. the denominator of which shall be the Market Price multiplied
     by the total number of shares of Common Stock outstanding immediately
     after the issuance of such additional shares of Common Stock.

     No adjustments of the Exercise Price shall be made under this paragraph (d)
upon the issuance of any additional shares of Common Stock that (y) are issued
pursuant to thrift plans, stock purchase plans, stock bonus plans, stock option
plans, employee stock ownership plans and other incentive or profit sharing
arrangements for the benefit of employees ("Employee Benefit Plans") that
otherwise would cause an adjustment under this paragraph (d); provided that the
aggregate number of shares of Common Stock so issued (including the shares
issued pursuant to any options, rights or warrants or convertible or
exchangeable securities issued under such Employee Benefit Plans containing the
right to purchase shares of Common Stock) pursuant to Employee Benefit Plans
after the closing date of the IPO, as adjusted for any stock splits, stock
dividends or subdivisions or combinations of Common Stock prior to the
Expiration Date, shall not in the aggregate exceed 5%


                                       -4-

<PAGE>   5
of the Company's outstanding Common Stock at the time of such issuance; or (z)
are issued pursuant to any Common Stock Equivalent (as hereinafter defined) (i)
if upon the issuance of any such Common Stock Equivalent, any such adjustments
shall previously have been made pursuant to paragraph (e) of this Section 2 or
(ii) if no adjustment was required pursuant to paragraph (e) of this Section 2.

          (e) Common Stock Equivalents. If the Company shall, after the Issuance
Date, issue any security or evidence of indebtedness which is convertible into
or exchangeable for Common Stock ("Convertible Security"), or any warrant,
option or other right to subscribe for or purchase Common Stock or any
Convertible Security, other than pursuant to Employee Benefit Plans (together
with Convertible Securities, "Common Stock Equivalent"), or if, after any such
issuance, the price per share for which additional shares of Common Stock may be
issuable thereunder is amended, then the Exercise Price upon each such issuance
or amendment shall be adjusted as provided in subsection (d) on the basis that
(i) the maximum number of additional shares of Common Stock issuable pursuant to
all such Common Stock Equivalents shall be deemed to have been issued as of the
earlier of (a) the date on which the Company shall enter into firm contract for
the issuance of such Common Stock Equivalent, or (b) the date of actual issuance
of such Common Stock Equivalent; and (ii) the aggregate consideration for such
maximum number of additional shares of Common Stock shall be deemed to be the
minimum consideration received and receivable by the Company for the issuance of
such additional shares of Common Stock pursuant to such Common Stock Equivalent;
provided however, that no adjustment shall be made pursuant to this subsection
(e) unless the consideration received and receivable by the Company per share of
Common Stock for the issuance of such additional shares of Common Stock pursuant
to such Common Stock Equivalent is less than the Market Price. No adjustment of
the Exercise Price shall be made under this subsection (e) upon the issuance of
any Convertible Security which is issued pursuant to the exercise of any
warrants or other subscription or purchase rights therefor, if any adjustment
shall previously have been made in the Exercise Price then in effect upon the
issuance of such warrants or other rights pursuant to this subsection (e).

          (f) Miscellaneous. The following provisions shall be applicable to the
making of adjustments in the Exercise Price hereinbefore provided in this
Section 2:

              i. The consideration received by the Company shall be deemed to be
     the following: (I) to the extent that any additional shares of Common
     Stock or any Common Stock Equivalent shall be issued for cash
     consideration, the consideration received by the Company therefor, or, if
     such additional shares of Common Stock or Common Stock Equivalent are
     offered by the Company for subscription, the subscription price, or, if
     such additional shares of Common Stock or Common Stock Equivalent are sold
     to underwriters or dealers for public offering without a subscription
     offering, the initial public offering price, in any such case excluding any
     amounts paid or receivable for accrued interest or accrued


                                       -5-
<PAGE>   6

     dividends and without deduction of any compensation, discounts, commissions
     or expenses paid or incurred by the Company for and in the underwriting of,
     or otherwise in connection with, the issue thereof, (II) to the extent that
     such issuance shall be for consideration other than cash, then, except as
     herein otherwise expressly provided, the fair value of such consideration
     at the time of such issuance as determined in good faith by the Board of
     Directors, as evidenced by certified resolution of the Board of Directors
     delivered to the holder of this Warrant Certificate setting forth such
     determination. The consideration for any additional shares of Common Stock
     issuable pursuant to any Common Stock Equivalent shall be the consideration
     received by the Company for issuing such Common Stock Equivalent, plus the
     additional consideration payable to the Company upon the exercise,
     conversion or exchange of such Common Stock Equivalent. In case of the
     issuance at any time of any additional shares of Common Stock or Common
     Stock Equivalent in payment or satisfaction of any dividend upon any class
     of stock other than Common Stock, the Company shall be deemed to have
     received for such additional shares of Common Stock or Common Stock
     Equivalent (which shall not be deemed to be dividend payable in, or other
     distribution of, Common Stock under subsection (c) above) consideration
     equal to the amount of such dividend so paid or satisfied.

              ii. Upon the expiration of the right to convert, exchange or
     exercise any Common Stock Equivalent the issuance of which effected an
     adjustment in the Exercise Price, if any such Common Stock Equivalent shall
     not have been converted, exercised or exchanged, the number of shares of
     Common Stock deemed to be issued and outstanding because they were issuable
     upon conversion, exchange or exercise of any such Common Stock Equivalent
     shall no longer be computed as set forth above, and the Exercise Price
     shall forthwith be readjusted and thereafter be the price which it would
     have been (but reflecting any other adjustments in the Exercise Price made
     pursuant to the provisions of subsection (d) after the issuance of such
     Common Stock Equivalent) had the adjustment of the Exercise Price made upon
     the issuance or sale of such Common Stock Equivalent been made on the basis
     of the issuance only of the number of additional shares of Common Stock
     actually issued upon exercise, conversion or exchange of such Common Stock
     Equivalent and thereupon only the number of additional shares of Common
     Stock actually so issued shall be deemed to have been issued and only the
     consideration actually received by the Company (computed as in subparagraph
     (i) of this paragraph (f)) shall be deemed to have been received by the
     Company.

              iii. The number of shares of Common Stock at any time outstanding
     shall not include any shares thereof then directly or indirectly owned or
     held by or for the account of the Company or its wholly owned subsidiaries.


                                       -6-

<PAGE>   7
              iv. For the purposes of this Section 2, the term "shares of Common
     Stock" shall mean shares of (i) the class of stock designated as the Common
     Stock at the date hereof or (ii) any other class of stock resulting from
     successive changes or reclassifications of such shares consisting solely of
     changes in par value, or from par value to no par value, or from no par
     value to par value. If at any time, because of an adjustment pursuant to
     subsection (a), the Warrants shall entitle the holders to purchase any
     securities other than shares of Common Stock, thereafter the number of such
     other securities so purchasable upon exercise of each Warrant and the
     Exercise Price of such securities shall be subject to adjustment from time
     to time in a manner and on terms as nearly equivalent as practicable to the
     provisions with respect to the Common Stock contained in this Section 2.

          (g) Calculation of Exercise Price. The Exercise Price in effect from
time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

     3. NOTICE OF ADJUSTMENTS. Whenever the Exercise Price or the number of
shares of Common Stock is required to be adjusted as provided in Section 2, the
Company shall forthwith compute the adjusted Exercise Price or the number of
shares of Common Stock issuable and shall prepare and mail to the holder hereof
a certificate setting forth such adjusted Exercise Price or such number of
shares of Common Stock, showing in reasonable detail the facts upon which the
adjustment is based.

     4. VOLUNTARY REDUCTION. (a) The Company may at its option, but shall not be
obligated to, at any time during the term of the Warrants, reduce the then
current Exercise Price by any amount selected by the Board of Directors;
provided that if the Company elects so to reduce the then current Exercise
Price, such reduction shall be irrevocable during its effective period and
remain in effect for minimum of 30 days following the date of such election,
after which time the Company may, at its option, reinstate the Exercise Price in
effect prior to such reduction. Whenever the Exercise Price is reduced, the
Company shall mail to the holder notice of the reduction at least 30 days before
the date the reduced Exercise Price takes effect, stating the reduced Exercise
Price and the period for which such reduced Exercise Price will be in effect.

          (b) The Company may make such decreases in the Exercise Price, in
addition to those required or allowed by this Section 4, as shall be determined
by it, as evidenced by certified resolution of the Board of Directors delivered
to the holders, to be advisable to avoid or diminish any income tax to the
holder resulting from any dividend or distribution of stock or issuance of
rights or warrants to purchase or subscribe for stock or from any event treated
as such for income tax purposes.


                                       -7-

<PAGE>   8

     5. NOTICES TO WARRANT HOLDERS. In the event:

          (a) the Company shall authorize any consolidation or merger to which
the Company is a party and for which approval of any stockholders of the Company
is required, or of the conveyance or sale of all or substantially all of the
assets of the Company, or of any reclassification or change of the Common Stock
or other securities issuable upon exercise of the Warrants (other than a change
in par value, or from par value to no par value, or from no par value to par
value or as a result of subdivision or combination), or a tender offer or
exchange offer for shares of Common Stock (or other securities issuable upon the
exercise of the Warrants); or

          (b) the Company shall declare any dividend (or any other distribution)
on the Common Stock or any other class of its capital stock; or

          (c) the Company shall authorize the granting to the holders of Common
Stock or any other class of its capital stock of rights or warrants to subscribe
for or purchase any shares of any class or series of capital stock or any other
securities convertible into or exchangeable for shares of stock; or

          (d) of the voluntary or involuntary dissolution, liquidation or
winding up of the Company;

     then the Company shall cause to be sent to the holder hereof, at least 30
days prior to the applicable record date hereinafter specified, or promptly in
the case of events for which there is no record date, a written notice stating
(x) the date for the determination of the holders of record of shares of Common
Stock (or other securities issuable upon the exercise of the Warrants) entitled
to receive any such dividends or other distribution, (y) the initial expiration
date set forth in any tender offer or exchange offer for shares of Common Stock
(or other securities issuable upon the exercise of the Warrants), or (z) the
date on which any of the events specified in subsections (a)-(d) is expected to
become effective or consummated, and the date as of which it is expected that
holders of record of shares of Common Stock (or other securities issuable upon
the exercise of the Warrants) shall be entitled to exchange such shares for
securities or other property, if any, deliverable upon any such event. Failure
to give such notice or any defect therein shall not affect the legality or
validity of any such event, or the vote upon any such action.

     6. REPORTS TO WARRANT HOLDERS. The Company will cause to be delivered, by
first-class mail, postage prepaid, to the holder at such holder's address
appearing hereon, or such other address as the holder shall specify, a copy of
any reports delivered by the Company to the holders of Common Stock.


                                       -8-

<PAGE>   9
     7. COVENANTS OF THE COMPANY. The Company covenants and agrees that:

          (a) Until the Expiration Date, the Company shall at all times reserve
and keep available, free from preemptive rights, out of the aggregate of its
authorized but unissued Common Stock (and other securities), for the purpose of
enabling it to satisfy any obligation to issue shares of Common Stock (and other
securities) upon the exercise of the Warrants, the number of shares of Common
Stock (and other securities) issuable upon the exercise of such Warrants.

          (b) The Company shall pay all expenses, taxes and other charges
payable in connection with the preparation, issuance and delivery of new warrant
certificates on transfer of the Warrants.

          (c) All Common Stock (and other securities) which may be issued upon
exercise of the Warrants shall upon issuance be validly issued, fully paid,
non-assessable and free from all preemptive rights and all taxes, liens and
charges with respect to the issuance thereof, and will not be subject to any
restrictions on voting or transfer thereof except as set forth in any
stockholders agreement.

          (d) All original issue taxes payable in respect of the issuance of
shares of Common Stock to the registered holder hereof upon the exercise of the
Warrants shall be borne by the Company; provided, that the Company shall not be
required to pay any tax or charge imposed in connection with any transfer
involved in the issuance of any certificate representing shares of Common Stock
(and other securities) in any name other than that of the registered holder
hereof, and in such case the Company shall not be required to issue or deliver
any certificate representing shares of Common Stock (and other securities) until
such tax or other charge has been paid or it has been established to the
Company's satisfaction that no such tax or charge is due.

          (e) As soon as practicable after the receipt from the holder of this
Warrant Certificate of notice of the exercise of number of warrants sufficient
to require filing under the Hart-Scott-Rodino Antitrust Improvements Act of 1976
and the rules, regulations and formal interpretations thereunder, as amended
from time to time (the "HSR Act"), but in any event no later than the 10th
business day after receipt of such notice, the Company will (i) prepare and
file with the Antitrust Division of the Department of Justice (the "DOJ") and
the Federal Trade Commission (the "FTC") the Notification and Report Form
(accompanied by all documentary attachments contemplated thereby) required by
the HSR Act, (ii) upon the request of the holder, request early termination of
the waiting period imposed by the HSR Act, (iii) coordinate and cooperate with
the holder in responding to formal and informal requests for additional
information and documentary material from the DOJ and the FTC in connection with
such filing, (iv) use its best efforts to take, or cause to be taken, all
reasonable action and to do, or cause to be done, all things necessary and
appropriate to permit the issuance to the holder of the shares of Common Stock
issuable upon the



                                      -9-
<PAGE>   10
exercise of the warrants with respect to which any filing is required under the
HSR Act, and (v) reimburse the holder for the entire amount of any filing fee or
any other costs and expenses incurred by the holder in connection therewith
(including legal fees), or as required to be paid under the HSR Act.

          (f) QUOTATION ON NASDAQ. The Company shall maintain the designation
and quotations, or listing, of its Common Stock on the NASDAQ national market
(or on the New York Stock Exchange or the American Stock Exchange) until the
date on which none of the Warrants or Warrant Shares remain outstanding.

     8. NO RIGHTS AS STOCKHOLDER. The holder of the Warrants shall not, by
virtue of holding such Warrants, be entitled to any rights of a stockholder of
the Company either at law or in equity, and the rights of the holder of the
Warrants are limited to those expressed herein.

     9. NOTICES. All notices provided for hereunder shall be in writing and may
be given by registered or certified mail, return receipt requested, telex,
telegram, telecopier, air courier guaranteeing overnight delivery of personal
delivery, if to the holder at the following address:

          Joint Energy Development Investments II
          Limited Partnership.
          1400 Smith Street
          Houston, Texas 77002
          Attention: Donna Lowry
          Telecopier: (713) 646-4039

and, if to the Company:

          Brigham Exploration Company
          6300 Bride Point Parkway
          Building 2; Suite 500
          Austin, Texas 78730
          Attention: Craig M. Fleming
          Telecopier: (512) 472-3400

     10. GOVERNING LAW. This Warrant Certificate shall be governed by and
construed in accordance with the laws of the State of Texas without regard to
principles of conflict of laws.

     11. LOST, STOLEN, MUTILATED OR DESTROYED WARRANT CERTIFICATES. Upon receipt
by the Company of evidence reasonably satisfactory to it of the ownership of and
the loss, theft, destruction or mutilation of any Warrant Certificate, then, in
the absence of notice to the Company that such


                                      -10-

<PAGE>   11
Warrant Certificate has been acquired by a bona fide purchaser, the Company
shall execute and deliver, in exchange for or in lieu of the lost, stolen,
destroyed or mutilated Warrant Certificate, a substitute Warrant Certificate of
the same tenor and evidencing a like number of Warrants.

     12. ASSIGNMENT. The holder of this Warrant Certificate shall be entitled,
without obtaining the consent of the Company, to transfer or assign its rights,
title and interest in (and rights, title and interest under) this Warrant
Certificate in whole or in part to any Person or Persons. Upon surrender of this
Warrant Certificate to the Company, with the Transfer Form annexed hereto as
Exhibit B duly executed, the Company shall, without charge, execute and deliver
a new warrant certificate or warrant certificates, identical in form to this
Warrant Certificate, evidencing the number of Warrants being transferred
pursuant to the Transfer Form in the name of the assignee or assignees named in
such Transfer Form. If the holder's entire interest is not being assigned, the
Company shall, without charge, execute and deliver one or more new warrant
certificates identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number being transferred pursuant to the
Transfer Form, and this Warrant Certificate shall promptly be canceled. The
terms and provisions of this Warrant Certificate shall inure to the benefit of
the holder and its successors and assigns and shall be binding upon the Company
and its successors and assigns, including, without limitation, any Person
succeeding the Company by merger, consolidation or acquisition of all or
substantially all of the Company's assets.



                                      -11-

<PAGE>   12
         IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to
be executed as of March 26, 1999, by the undersigned, thereunto duly authorized.

                                        BRIGHAM EXPLORATION COMPANY

                                        By:
                                           ---------------------------------
                                            Karen E. Lynch
                                            Vice President



                                      -12-

<PAGE>   13
                                    EXHIBIT A

                              ELECTION TO EXERCISE
          [To be executed on exercise of the Warrants evidenced by this
                              Warrant Certificate]

TO:       Brigham Exploration Company

          The undersigned, the holder of the Warrants evidenced by the attached
Warrant Certificate, hereby irrevocably elects to exercise Warrants, and
herewith makes payment of ________________________ ($____________) representing
the aggregate Exercise Price thereof, and requests that the certificate
representing the securities issuable hereunder be issued in the name of
______________________ and delivered to _______________________, whose address 
is __________________________.

          Dated:
                ------------------------

                    Name of Registered Holder:
                                              ---------------------------------
                    Signature:
                              -------------------------------------------------
                    Title:
                          -----------------------------------------------------
                    Address:
                            ---------------------------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.


                                      -13-

<PAGE>   14
                                    EXHIBIT B

                                  TRANSFER FORM
      [To be executed only upon transfer of the Warrants evidenced by this
                              Warrant Certificate]

          FOR VALUE RECEIVED, the undersigned hereby sells, assigns and
transfers unto __________________________ the Warrants represented by the
within Warrant Certificate, together with all right, title and interest therein,
and does hereby irrevocably constitute and appoint ____________________________
Attorney-in-Fact, to transfer same on the books of the Company with full power
of substitution in the premises.

          Dated:
                ------------------------

                    Name of Registered Holder:
                                              ---------------------------------
                    Signature:
                              -------------------------------------------------
                    Title:
                          -----------------------------------------------------
                    Address:
                            ---------------------------------------------------


WITNESS:

- ----------------------------------------


NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.




                                      -14-


<PAGE>   1
                                                                   EXHIBIT (iii)

                   AMENDMENT TO REGISTRATION RIGHTS AGREEMENT

          This AMENDMENT TO REGISTRATION RIGHTS AGREEMENT (the "Amendment")
dated as of March 26, 1999, among Brigham Exploration Company, Delaware
corporation (the "Company"), Joint Energy Development Investments II Limited
Partnership, a Delaware limited partnership ("JEDI-II"), and ECT Merchant
Investments Corp, ("ECT Merchant"), amends the Registration Rights Agreement
dated as of August 20, 1998 among the Company, JEDI-II and Enron Capital &
Trade Resources Corp., a Delaware corporation ("ECT").

          NOW, THEREFORE, in consideration of the mutual covenants and
agreements set forth herein and for good and valuable consideration, the receipt
of which is hereby acknowledged, the parties agree as follows:

                                    ARTICLE I
            MODIFICATION OF DEFINITION OF WARRANTS AND WARRANT SHARES

          Section 1.01. Definition of Warrant. "Warrants" means the Warrants
issued by the Company to JEDI-II and ECT Merchant as of the date hereof for the
purchase of an aggregate of 1,000,000 shares of Common Stock, evidenced as of
the date hereof by Warrant Certificates Nos. A-3 and A-4, and any Warrants
issued upon the transfer thereof or in substitution therefor, as the same may be
amended from time to time.

          Section 1.02. Definition of Warrant Shares. "Warrant Shares" shall
mean the shares of Common Stock and other securities receivable upon exercise
of the Warrants.

                                   ARTICLE II
                                  MISCELLANEOUS

          Section 2.01 Execution in Counterparts. This Amendment may be
executed in any number of counterparts and by different parties on separate
counterparts and all of such counterparts shall together constitute one and the
same instrument.

          Section 2.02 Governing Law. This Amendment and the rights and
obligations of the parties hereunder shall be construed in accordance with and
be governed by the laws of the State of Texas and the United States of America.

          Section 2.01 Descriptive Headings Etc. The descriptive headings of the
sections of this Amendment are inserted for convenience only and shall not be
deemed to affect the meaning or construction of any of the provisions hereof. 



<PAGE>   2

          IN WITNESS WHEREOF, the parties have executed this Amendment to
Registration Rights Agreement as of the date first written above.

                              BRIGHAM EXPLORATION COMPANY

                              By:
                                 ------------------------------
                                   Karen E. Lynch
                                   Vice President

                              JOINT ENERGY DEVELOPMENT
                              INVESTMENTS II LIMITED PARTNERSHIP

                              By:  Enron Capital Management II Limited
                                   Partnership, its General Partner

                                   By: Enron Capital 11 Corp., its General
                                       Partner

                                   By:
                                      ----------------------------------------
                                        Mark J. Warner
                                        Agent and Attorney-in-fact

                              ECT MERCHANT INVESTMENTS CORP.

                              By:
                                      -------------------------------------
                              Name: 
                                      -------------------------------------
                              Title:
                                      -------------------------------------




                                       -2-



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