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                                                                 Exhibit 4.2.1

                          FIRST AMENDMENT TO INDENTURE


         This FIRST AMENDMENT TO INDENTURE, dated as of March 26, 1999 ("this
Amendment"), is among BRIGHAM EXPLORATION COMPANY, a corporation duly organized
and existing under the laws of the State of Delaware (the "Borrower"), and
CHASE BANK OF TEXAS, NATIONAL ASSOCIATION, a national banking association
existing under the laws of the United States, as trustee (the "Trustee").

                             PRELIMINARY STATEMENT

         All covenants and agreement made by the Borrower herein are for the
benefit and security of the holders of the Borrower's Senior Subordinated
Secured Notes due 2003.

                                    RECITALS

         The Borrower and the Trustee are parties to that certain Indenture
dated as of August 20, 1998 (the "Indenture"). Capitalized terms used and not
otherwise defined herein are used with the meanings ascribed thereto in the
Indenture.

         The Borrower has advised the Noteholders and the Trustee that it
desires to amend certain provisions of the Indenture, and the Borrower has
requested that the Trustee and the Noteholders agree to various amendments to
certain provisions of the Indenture.

         The Trustee, upon the consent and authorization of the Noteholders,
has agreed to so amend certain provisions of the Indenture upon the terms and
subject to the conditions and limitations of this Amendment.

         NOW, THEREFORE, in consideration of the premises, covenants and
agreements contained herein, the parties hereto agrees as follows:

         Section 1. Definitions. The following capitalized terms shall have the
following respective meanings when used herein:

         A. "Duke Consent Letter" shall mean that certain Letter Agreement
dated as of February 18, 1999 executed by the Borrower and the Trustee.

         B. "Duke Transaction" shall mean the transactions contemplated by the
Duke Consent Letter.

         C. "Lending Relationship" shall refer to the Indenture and the other
Loan Documents, including, without limitation, this Amendment, together with any
and all negotiations, discussions, acts, omissions, renewals, extensions, and
other agreements or events related to the Indenture and such other Loan
Documents, the parties' obligations thereunder and the transactions contemplated



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thereby, including, without limitation, any such negotiations, discussions,
acts, omissions, renewals, extensions, other agreements or events that (a)
occurred prior to the date hereof, (b) may occur on the date hereof, or (c)
occurred prior to the execution of this Amendment and the instruments and
documents executed and delivered in connection herewith or relating hereto.

         D. "New Mortgage" shall mean that certain Mortgage, Deed of Trust,
Assignment of Production, Security Agreement and Financing Statement
(Non-Producing Properties) dated effective as of the date hereof from Brigham
Oil & Gas, L.P. to Gray H. Muzzy, as Trustee, for the benefit of Trustee.

         E. "RELEASED CLAIMS" SHALL MEAN ANY AND ALL CLAIMS (INCLUDING WITHOUT
LIMITATION ANY LIABILITIES, DAMAGES, DEMANDS AND CAUSES OF ACTION ARISING
THEREFROM), WHETHER (A) AT LAW OR IN EQUITY, (B) ON THE ALLEGED COMMISSION OF A
TORT, (C) ON THE ALLEGED BREACH (OR ANTICIPATORY BREACH OR REPUDIATION) OF ANY
CONTRACT, DUTY, OR WARRANTY (WHETHER ORAL OR WRITTEN, EXPRESS OR IMPLIED), (D)
ON THE ALLEGED VIOLATION OF ANY STATUTE, TARIFF, OR REGULATION (WHETHER
PROMULGATED BY THE UNITED STATES, ANY STATE THEREOF, ANY FOREIGN STATE OR
COUNTRY, OR ANY OTHER GOVERNMENTAL AGENCY OR ENTITY, WHEREVER LOCATED), OR (E)
ON ANY OTHER FACTUAL, LEGAL OR EQUITABLE THEORY, INCLUDING, WITHOUT LIMITATION,
ANY CLAIM FOR DAMAGES OF ANY TYPE OR NATURE, FOR INJUNCTIVE OR OTHER RELIEF,
FOR ATTORNEYS' FEES, INTEREST OR ANY OTHER LIABILITY WHATSOEVER ON ANY THEORY,
INCLUDING WITHOUT LIMITATION ANY LOSS, COST OR DAMAGE IN CONNECTION WITH OR
BASED UPON "LENDER LIABILITY", UNFAIR DEALING, DURESS, COERCION, CONTROL OR
UNDUE INFLUENCE, EXTORTION OR COMMERCIAL BRIBERY, BREACH OF AN IMPLIED COVENANT
OR DUTY OF GOOD FAITH AND FAIR DEALING, MATERIAL MISREPRESENTATION OR OMISSION,
OVERREACHING, UNCONSCIONABILITY, CONFLICT OF INTEREST, BAD FAITH, MALPRACTICE,
DISPARATE BARGAINING POSITION, DETRIMENTAL RELIANCE, PROMISSORY ESTOPPEL,
ESTOPPEL BY DEED, WAIVER, LACHES, OR ANY OTHER EQUITABLE THEORY, EQUITABLE
SUBORDINATION, BREACH OF FIDUCIARY DUTY OR ANY OTHER DUTY, OR TORTIOUS
INDUCEMENT TO COMMIT SUCH BREACH, TORTIOUS INTERFERENCE WITH CONTRACT OR
PROSPECTIVE BUSINESS RELATIONS, NEGLIGENT PERFORMANCE OF CONTRACTUAL
OBLIGATIONS, OR OTHER THEORIES OF NEGLIGENCE, NEGLIGENT OR INTENTIONAL
INFLICTION OF EMOTIONAL DISTRESS, SLANDER, LIBEL, OTHER DEFAMATION, FRAUDULENT
TRANSFER, CONVERSION, TRESPASS TO (OR CLOUDING THE TITLE OF) PROPERTY, USURY,
VIOLATIONS OF THE RACKETEER INFLUENCED AND CORRUPT ORGANIZATIONS ACT, DECEPTIVE
TRADE PRACTICES, CONSPIRACY, OR ANY THEORY OF LIABILITY AS PARTNERS OR JOINT
VENTURERS, THAT ANY RELEASING PARTY MAY HAVE AS OF THE DATE HEREOF AGAINST ANY
RELEASED PARTY WITH RESPECT TO THE LENDING RELATIONSHIP.

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         F. "Released Party" shall mean each of the Trustee, the Noteholders
and their respective predecessors, successors, assigns, directors, officers,
partners, employees, agents, attorneys, principals and Affiliates and all other
Persons liable or who might be claimed to be liable on their behalf
(collectively, the "Released Parties").

         G. "Releasing Party" shall mean each of the Borrower and the
Guarantors and their respective predecessors, successors, assigns, directors,
officers, partners, employees, agents, attorneys, principals, Affiliates and
all other Persons who might have a claim against any Released Party
(collectively, the "Releasing Parties").

Section 2.  Amendments to Indenture.  The Indenture is amended hereby as 
follows:

         A. Section 1.01 is amended hereby as follows:

                  (i) by deleting the definition of the term "Basic Documents"
in its entirety and substituting the following therefor:

                  ""Basic Documents" means, collectively, this Indenture, the
         Securities Purchase Agreement and the other Loan Documents, as each
         may be amended, supplemented or restated from time to time.";

                  (ii) by inserting the following new definitions:

                  ""First Borrowing Base Determination Date" means June 1,
1999.";

                  ""Measurement Date" has the meaning set forth in the last
paragraph of Section 9.02(a)"

                  (iii) by inserting in the definition of the term "Guarantors"
after the reference to "Brigham Holdings II, LLC", the reference "Sooner
Production LLC"; and

                  (iv) by deleting the reference "Section 7.09" in the
definition of the term "Mortgage" and substituting the reference "Section
7.09(a)" therefor.

         B. Section 4.01(a) of the Indenture is amended hereby as follows:

                  (i) by inserting in clause (ii) following the reference "and
BOG", the phrase ", Sooner Production LLC and Quest Resources LLC"; and

                  (ii) by inserting at the end of Section 4.01(a) the
following:

                  "; and (v) a Security Agreement substantially in the form of
         Exhibit D duly executed by BOG in favor of the Trustee for the ratable
         benefit of the Noteholders granting a security interest in all of
         BOG's right, title and interest in and to the Capital Stock of DND Oil
         & Gas, L.P. and Quest Resources LLC, together with proper UCC-1
         Financing Statements duly filed

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         in Texas; and (vi) a Security Agreement substantially in the form of
         Exhibit D duly executed by Sooner Production LLC in favor of the
         Trustee for the ratable benefit of the Noteholders granting a security
         interest in all of BOG's right, title and interest in and to the
         Capital Stock of Quest Resources LLC, together with proper UCC-1
         Financing Statements duly filed in Texas, in each case as such
         Security Agreements may be amended, modified or supplemented from time
         to time."

         C. Section 4.01(c) of the Indenture is amended hereby as follows:

                  (i) by deleting the word "A" from the beginning of such
section and inserting the phrase "One or more".

         D. Section 4.01 (d) of the Indenture is amended hereby as follows:

                  (i) by deleting the words "BOG, Brigham, Inc., Brigham
Holdings I, LLC and Brigham Holding II, LLC" and replacing them with the words
"the Guarantors".

         E. Section 7.01 of the Indenture is amended hereby as follows:

                  (i) by inserting the following new subsection (c):

                  "(c) Monthly Financial Statements. As soon as available and
         in any event within thirty (30) days after the end of each calendar
         month that is not also the end of one of the Borrower's first three
         fiscal quarterly periods or of the Borrower's fiscal year,
         consolidated statements of income and changes in financial position of
         the Borrower and its Consolidated Subsidiaries for such period and for
         the period from the beginning of the respective fiscal year to the end
         of such period, and the related consolidated balance sheets as at the
         end of such period and, beginning March 31, 2000, statements setting
         forth in each case in comparative form the corresponding figures for
         the corresponding period in the preceding fiscal year, accompanied by
         the certificate of a Responsible Officer, which certificate shall
         state that such financial statements fairly present the consolidated
         financial condition and results of operations of the Borrower and its
         Consolidated Subsidiaries in accordance with GAAP, as at the end of,
         and for, such period (subject to normal year-end audit adjustments).";

                  (ii) by relettering the existing subsections (c) through (i)
as subsections (d) through (j);

                  (iii) by deleting in subsection (i), as relettered by (ii)
above, the reference "Concurrent with the First Reserve Report and each January
1 Reserve Report thereafter" and substituting therefor the reference "On or
before January 31 of each year"; and

                  (iv) by inserting after the reference "paragraph (a) or (b)"
in the final paragraph of Section 7.01 the reference "or (c)" and by inserting
the phrase "(and for each June 30 quarter end

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and December 31 year end, commencing June 30, 2000, calculating the ratio
described in the last paragraph of Section 9.02(a) for such Measurement Date).

         F. Section 7.07 of the Indenture is amended hereby by deleting each
reference to "January 1" in subsection (b) and substituting therefor the
reference "January 31".


         G. Section 7.09(a) of the Indenture is amended to add the following
proviso at the end thereof:

         "; provided that, with respect to any non-proved properties, the
         phrase "second-priority" as found in the fifth line of this subsection
         7.09(a) shall be replaced with the word "perfected" and the phrase
         "previous Collateral Documents" as found in the twelfth line of
         subsection 7.09(a) shall be replaced with the phrase "New Mortgage and
         any associated financing statements"."

         H. Section 8.01 of the Indenture is amended hereby as follows:

                  (i) by deleting the reference "1.5 to 1.0" in the first
sentence of subsection (a) and substituting therefore the reference "(A) for
any Debt incurred on or after July 1, 1999 and on or before December 31, 1999,
1.3 to 1.0 and (B) for any Debt incurred on on or after January 1, 2000, 1.5 to
1.0."; and

                  (ii) by deleting the sum "$75,000,000," from the last
sentence of subsection (a) and replacing it with the phrase "the lesser of (i)
$75,000,000 and (ii) the loan commitments available from time to time under
such refinanced Senior Loan,".

         I. Section 8.16 of the Indenture is amended hereby deleting the
section in its entirety and substituting the following therefor:

         "Section 8.16 Consolidated Interest Coverage Ratio. As of the last day
         of each fiscal quarter, beginning December 31, 1999, the Borrower will
         not permit the Consolidated Interest Coverage Ratio to be less than
         (i) 1.5 to 1.0 as of December 31, 1999, (ii) 1.75 to 1.0 as of March
         31, 2000, and (iii) 2.0 to 1.0 as of the end of each fiscal quarter
         thereafter."

         J. Section 8.17 of the Indenture is amended hereby by deleting the
reference "The" from the beginning of the first sentence and substituting
therefore the reference "On and after December 31, 1999, the".

         K. Section 9.02 of the Indenture is amended hereby as follows:

                  (i) by adding to the end of subsection (a) the following:

         "Notwithstanding the foregoing (i) if, as of any June 30 or December
         31, commencing with June 30, 2000 (each such date being a "Measurement
         Date"), the ratio of (A) the Borrower's


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         Adjusted Consolidated Net Tangible Assets as of such Measurement Date
         to (B) the sum of (1) Borrower's Consolidated Indebtedness as of such
         Measurement Date plus (2) past due interest on Debt as of such
         Measurement Date, is less than 2.0 to 1.0, the applicable rate of
         interest specified in this Section 9.01(a) as then in effect
         (including after giving effect to increases that may have occurred
         prior to the applicable Measurement Date, if any) shall increase by
         0.5% effective as of such Measurement Date. Any such increase shall be
         effective from the applicable Measurement Date and remain effective
         (subject to subsequent additional increases pursuant to this
         paragraph, if any) until the Obligations are paid in full, subject to
         the provisions of Section 9.02(c) and 9.02(f)."; and

                  (ii) by deleting the first five (5) sentences of subsection
(b) and substituting the following therefore:

         "The Borrower shall have the option to pay accrued interest on the
         Notes in kind on each of the six (6) consecutive Interest Payment
         Dates commencing with and following February 20, 1999, as provided in
         this Section 9.02(b). In such event the accrued interest due on such
         Interest Payment Dates shall be calculated at the rates set forth in
         this Section 9.02(b) and the interest due (calculated at the rates set
         forth in this Section 9.02(b)) shall be deemed an advance of principal
         on the Notes and, as of the applicable Interest Payment Date, shall be
         added to the outstanding principal balance of the Notes
         (notwithstanding the outstanding principal balance may exceed, in the
         aggregate, the face amount of the Notes). In order to not exercise its
         option under this Section 9.02(b), the Borrower must, on or before the
         applicable Interest Payment Date, deliver written notice to the Agent
         executed by a Responsible Officer notifying Agent of its election to
         not pay interest in kind. Should Borrower fail to deliver such written
         notice in a timely fashion, Borrower shall be deemed to have
         irrevocably elected to make payment of accrued interest in kind on the
         next applicable Interest Payment Date. Should Borrower deliver such
         notice in a timely fashion, Borrower shall be deemed to have
         irrevocably elected to make payment of accrued interest in cash and
         any subsequent failure to do so in a timely fashion (subject to the
         thirty (30) day grace period provided in Section 10.01(a)) shall
         constitute an Event of Default hereunder."

         L. Exhibit D to the Indenture is amended in its entirety to read as
set forth in Exhibit D attached hereto.

         Section 3. Covenants. The Borrower covenants and agrees that during
the period from February 1, 1999 through and including June 1, 1999:

         A. The Borrower shall deliver weekly cash budgets reasonably
satisfactory to the Agent in the form attached hereto as Exhibit B, and weekly
cash flow statements reasonably satisfactory to the Agent based on such form,
with variance analysis to budget (including accounts receivables and accounts
payables reporting) not later than the Friday following the week to which such
budgets and statements relate.

         B. The Borrower shall provide to the Agent from time to time upon
request by the Agent the certificate of a Responsible Officer of the Borrower
stating that, except as disclosed in a schedule


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thereto, the Borrower has not received written notice that any mechanics' liens
have been filed or will be filed on the Mortgaged Properties; provided that
mere receipt of an invoice for services rendered shall not constitute written
notice that a mechanics' lien will be filed.

         C. The Borrower shall cause its Subsidiary, DND Oil & Gas, LLP, to
transfer all assets held by it to BOG and, thereafter, to dissolve on or before
June 1, 1999.

         D. The Borrower will not, and will not allow any of its Subsidiaries
to, (i) transfer any assets to Quest Resources LLC or (ii) make any investments
in or loans or advances to Quest Resources LLC.

         E. The Borrower shall deliver within 30 days after the date of this
Agreement a First Amendment to Security Agreement executed by Brigham Holdings
I, LLC and a First Amendment to Security Agreement executed by Brigham Holdings
II, LLC, in each case in favor of the Agent, together with (a) evidence that
all other actions necessary or, in the opinion of the Agent, desirable to
perfect and protect the security interests and Liens granted thereby have been
taken, and (b) an opinion of the Borrower's counsel in form reasonably
satisfactory to the Agent.

Breach by the Borrower and its Subsidiaries of any of the foregoing covenants
(regardless of the reason therefor) shall constitute an Event of Default under
the Indenture.

         Section 4. Conditions Precedent. This Amendment shall become binding
upon receipt by the Agent of the following documents and satisfaction of the
other conditions provided in this Section 4, each of which must be satisfactory
to the Agent in form and substance:

         A. counterparts of this Amendment executed by the Borrower and the
Trustee;

         B. The Borrower shall have satisfied all of its obligations under
Section 7.09(a) of the Indenture, including the delivery to the Agent of all
Mortgages required thereby;

         C. evidence that all new Mortgages (including the New Mortgage) and
the related UCC-1 financing statements delivered pursuant to Section 7.09(a) of
the Indenture have been filed in all jurisdictions necessary to perfect and
protect the security interests, assignments and Liens created thereby;

         D. certificates of the Secretary or an Assistant Secretary of the
Borrower and each of the Guarantors setting forth for each of them (i) the
resolutions of its board of directors or managers (or if such Guarantor is a
partnership, resolutions of the general partner of such partnership), as
applicable, with respect to the authorization to execute and deliver this
Amendment and consummate the transactions contemplated hereby; (ii) the
Responsible Officer of such entity authorized to sign this Amendment, and (iii)
the signature of such authorized Responsible Officer of such entity;

         E. a monthly cash budget for 1999 of the Borrower and its
Subsidiaries, which shall include projected revenues, expenses and capital
expenditures;



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         F. a First Amendment to Security Agreement executed by the Borrower in
favor of the Trustee and evidence that all other actions necessary or, in the
opinion of the Agent desirable to perfect and protect the security interests
and Liens granted thereby have been taken;

         G. a First Amendment to Security Agreement executed by Brigham, Inc.
in favor of the Trustee granting, among other things, a second priority
security interest in all of Brigham, Inc.'s right, title and interest in its
ownership interests in Quest Resources LLC, and evidence that all other actions
necessary or, in the opinion of the Agent, desirable to perfect and protect the
security interests and Liens granted thereby have been taken;

         H. a First Amendment to Security Agreements executed by BOG in favor
of the Trustee granting, among other things, a second priority security
interest in all of BOG's right, title and interest in its ownership interests
in Quest Resources LLC, and evidence that all other actions necessary or, in
the opinion of the Agent, desirable to perfect and protect the security
interests and Liens granted thereby have been taken;

         I. a First Amendment to Security Agreement executed by Brigham
Holdings I, LLC in favor of the Trustee and evidence that all other actions
necessary or, in the opinion of the Agent, desirable to perfect and protect the
security interests and Liens granted thereby have been taken;

         J. a First Amendment to Security Agreement executed by Brigham
Holdings II, LLC in favor of the Trustee and evidence that all other actions
necessary or, in the opinion of the Agent, desirable to perfect and protect the
security interests and Liens granted thereby have been taken;

         K. a First Amendment to Security Agreement executed by Sooner
Production LLC in favor of the Trustee and evidence that all other actions
necessary or, in the opinion of the Agent, desirable to perfect and protect the
security interests and Liens granted thereby have been taken;

         L. a Consent and Acknowledgment executed by each of the Guarantors;

         M. an opinion of counsel to Borrower substantially in the form
attached hereto as Exhibit C;

         N. duly executed warrants issued by the Borrower to ECT and JEDI-II in
their respective Participations in substitution for the original Warrants
issued pursuant to Section 2.03 of the Securities Purchase Agreement, for the
purchase of an aggregate of 1,000,000 shares of Common Stock, and which
warrants shall constitute the "Warrants" under the Indenture and the Securities
Purchase Agreement for all purposes;

         O. payment of the expenses of the Agent and the Noteholders in
accordance with Section 7 hereof; and

         P. such other documents as Agent or its counsel may reasonably
request.



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<PAGE>   9




         Section 5. Representations and Warranties.

         A. Except as provided in subsection (iii) of this Section 5.A., the
Borrower hereby reaffirms that, as of the date of this Amendment, the
representations and warranties made by the Borrower in the Securities Purchase
Agreement are true and correct as though made on and as of the date hereof, and
further, the Borrower represents that,

                  (i) as of the date hereof, no Default or Material Adverse
Effect has occurred and is continuing except as previously disclosed to the
Agent in writing;

                  (ii) the execution, delivery and performance by the Borrower
or the Guarantors of this Amendment and the other Loan Documents and all
instruments and documents to be delivered by the Borrower or the Guarantors, to
the extent a party thereto, hereunder and thereunder and the creation of all
Liens provided for herein and therein: (a) are within the Borrower's or such
Guarantor's corporate power; (b) have been duly authorized by all necessary or
proper corporate action, including the consent of stockholders, members and/or
partners therein or thereof; (c) are not in contravention of any provision of
the Borrower's or such Guarantor's certificate of incorporation, bylaws or
similar organizational and/or governing documents; (d) will not violate (1) any
law or regulation or (2) any order or decree of any court or governmental
instrumentality; (e) will not conflict with or result in the breach or
termination of, constitute a default under or accelerate any performance
required by, any indenture, mortgage, deed of trust, lease, agreement or other
instrument to which the Borrower or any of the Guarantors is a party or by
which the Borrower or any of the Guarantors or any of their respective property
is bound; (f) will not result in the creation or imposition of any Lien upon
any of the property of the Borrower or the Guarantors other than those in favor
of the Agent pursuant to the terms of this Amendment and the other Loan
Documents to be delivered in connection herewith; and (g) do not require the
consent or approval of any governmental body, agency, authority or any other
Person that has not been duly obtained, made or complied with prior to the date
hereof. At or prior to the date hereof, each of this Amendment and the other
Loan Documents to be delivered in connection herewith shall have been duly
executed and delivered for the benefit of or on behalf of the Borrower or the
Guarantors, in each case to the extent a party thereto, and each shall then
constitute a legal, valid and binding obligation of the Borrower or such
Guarantor, enforceable against it in accordance with its terms; and

                  (iii) notwithstanding the foregoing, the representations and
warranties contained in the last sentence of Section 4.10(a) of the Securities
Purchase Agreement (and not those contained in the first two sentences) are
reaffirmed with respect to the Mortgaged Property covered by or described in
the New Mortgage.

         B. Each of the Borrower and the Guarantors further represents and
warrants, for itself only that it (i) is executing this Amendment, and the
documents executed in connection herewith to which it is a party, after
consultation with counsel of its own choosing, (ii) has read and understands
the release granted by Section 6 hereof, (iii) desires to execute this
Amendment and such documents to which it is a party and (iv) has the requisite
authority to enter into and be bound by this Amendment and such documents to
which it is a party, including the release granted by Section 6 hereof.


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<PAGE>   10




         Section 6. Release.

         A. EACH OF THE RELEASING PARTIES DESIRES AND INTENDS FULLY TO
COMPROMISE, RELEASE AND SETTLE ANY AND ALL OF THE RELEASED CLAIMS; AND EACH OF
THE RELEASING PARTIES HEREBY COVENANTS, WARRANTS AND REPRESENTS UNTO EACH OF
THE RELEASED PARTIES THAT SUCH RELEASING PARTY DOES HEREBY FOREVER RELEASE,
ACQUIT, WAIVE AND DISCHARGE EACH OF THE RELEASED PARTIES OF AND FROM THE
RELEASED CLAIMS AND EACH OF THE RELEASING PARTIES HEREBY DECLARES THE SAME
FOREVER RELEASED, ACQUITTED, WAIVED, SETTLED AND DISCHARGED. THIS RELEASE IS
EFFECTIVE WITHOUT REGARD TO WHETHER (I) SUCH RELEASED CLAIMS ARE KNOWN OR
UNKNOWN, (II) DAMAGES ARISING OUT OF SUCH RELEASED CLAIMS HAVE YET ACCRUED,
(III) SUCH RELEASED CLAIMS AROSE COLLATERALLY, DIRECTLY, DERIVATIVELY, OR
OTHERWISE BETWEEN THE PARTIES HERETO OR (IV) AN ORDINARY PERSON IN THE SAME OR
SIMILAR CIRCUMSTANCES WOULD OR WOULD NOT, THROUGH THE EXERCISE OF DUE CARE,
HAVE DISCOVERED SUCH CLAIMS BY THE DATE OF THIS AMENDMENT. IN CONNECTION WITH
THE FOREGOING RELEASE:

         B. Borrower and each of the Guarantors represents and warrants that it
has the full power and authority to perform the release granted in this Section
6 and that it has not in any manner made any assignment of any Released Claim
to any third party.

         C. The release granted in this Section 6 will be effective upon
execution of this Amendment by all of the parties hereto.

         D. Each party executing this Amendment understands and agrees that the
release granted in this Section 6 is a full, final and complete release of the
Released Claims and that such release may be pleaded as an absolute and final
bar to any or all suits which may hereafter be filed or prosecuted by any one
or more of the Releasing Parties or anyone claiming by, through or under any
one or more of the Releasing Parties in respect of any of the matters released
hereby, and that no recovery on account of the Released Claims may hereafter be
had from any of the Released Parties; and that the consideration given for such
release is not an admission of liability or fault on the part of any of the
Released Parties (it being the express intent of the parties hereto to obtain
peace of mind and avoid the expense and uncertainty of potential litigation),
and that none of the Releasing Parties or those claiming by, through or under
any of them will ever claim that it is.

         E. The parties hereto acknowledge that the release granted by this
Section 6 does not have any effect with respect to relationships between the
Borrower and each of the Guarantors and the Noteholders and the Agent other
than in connection with the Lending Relationship.

         Section 7. Payment of Fees and Expenses; Form of Payment.

         A. The Borrower agrees, whether or not the transactions contemplated
hereby are consummated, to pay all reasonable expenses of the Agent and the
Noteholders (including, without


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<PAGE>   11




limitation, all reasonable fees and disbursements of counsel and other outside
consultants for the Agent and/or the Noteholders) in connection with the
negotiation, investigation, preparation, execution and delivery of, recording
and filing of, preservation of rights under and enforcement of this Amendment
and the other Loan Documents to be delivered in connection herewith.

         B. All payments to be made by the Borrower under this Amendment shall
be made in Dollars, in immediately available funds, to the Agent at such
account as the Agent shall specify.

          Section 8. Limitations. The amendments set forth herein are limited
precisely as written and shall not be deemed to (a) be a consent to, or waiver
or modification of, any other term or condition of the Securities Purchase
Agreement or the Indenture or any of the other Loan Documents, or (b) prejudice
any right or rights which the Noteholders or the Agent may now have or may have
in the future under or in connection with the Securities Purchase Agreement or
the Indenture or any of the other Loan Documents. Except as expressly
supplemented, amended or modified hereby, the terms and provisions of the
Securities Purchase Agreement or the Indenture or any other Loan Documents are
and shall remain in full force and effect. In the event of a conflict between
this Amendment and any of the foregoing documents, the terms of this Amendment
shall be controlling.

         Section 9. Governing Law. This Amendment and the rights and
obligations of the parties hereunder and under the Indenture shall be construed
in accordance with and be governed by the laws of the State of Texas and the
United States of America.

         Section 10. Descriptive Headings, etc. The descriptive headings of the
several 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.

         Section 11. 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.




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<PAGE>   12




         IN WITNESS WHEREOF, the parties hereto have caused this Amendment to
be executed as of the date first written above.

            NOTICE PURSUANT TO TEX. BUS. & COMM. CODE SECTION 26.02

THIS AMENDMENT AND OTHER LOAN DOCUMENTS EXECUTED BY ANY OF THE PARTIES BEFORE
OR SUBSTANTIALLY CONTEMPORANEOUSLY WITH THE EXECUTION HEREOF TOGETHER
CONSTITUTE A WRITTEN LOAN AGREEMENT AND REPRESENT THE FINAL AGREEMENT BETWEEN
THE PARTIES AND MAY NOT BE CONTRADICTED BY EVIDENCE OF PRIOR, CONTEMPORANEOUS
OR SUBSEQUENT ORAL AGREEMENTS OF THE PARTIES. THERE ARE NO UNWRITTEN ORAL
AGREEMENT BETWEEN THE PARTIES.


BORROWER:                           BRIGHAM EXPLORATION COMPANY


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



TRUSTEE:                            CHASE BANK OF TEXAS, NATIONAL ASSOCIATION,
                                    as Trustee


                                    By: /s/ Mauri J. Cowen                     
                                       ----------------------------------------
                                            Mauri J. Cowen
                                            Vice President and Trust Officer




                                      -12-

<PAGE>   13




GUARANTORS:                         BRIGHAM INC., a Nevada corporation


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


                                    BRIGHAM OIL & GAS, L.P.

                                    By: Brigham, Inc. a Nevada corporation, its
                                        General Partner


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


                                    BRIGHAM HOLDINGS I, LLC


                                    By: /s/ Ben M. Brigham 
                                       ----------------------------------------
                                    Name:  Ben M. Brigham 
                                         --------------------------------------
                                    Title: President 
                                           ------------------------------------


                                    BRIGHAM HOLDINGS II, LLC


                                    By: /s/ Ben M. Brigham 
                                       ----------------------------------------
                                    Name:  Ben M. Brigham  
                                         --------------------------------------
                                    Title: President 
                                          -------------------------------------


                                    DND OIL & GAS, L.P., a Texas limited 
                                    partnership

                                    By: Sooner Production Company, LLC, a Texas
                                        limited liability company, its General 
                                        Partner


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



                                      -13-

<PAGE>   14




                                    SOONER PRODUCTION COMPANY, LLC, a Texas
                                    limited liability company


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




                                      -14-

<PAGE>   15



The Noteholders have joined herein solely for the purpose of evidencing their
(i) consent to the foregoing, and (ii) authorization to the Trustee to execute
this Amendment:

                                    JOINT ENERGY DEVELOPMENT INVESTMENTS
                                    II LIMITED PARTNERSHIP, a Delaware limited
                                    partnership, as Purchaser

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

                                        By: Enron Capital II Corp., its
                                            General Partner


                                            By: /s/ Mark J. Warner            
                                                -------------------------------
                                                    Mark J. Warner
                                                    Agent and Attorney-in-Fact


                                    ENRON CAPITAL & TRADE RESOURCES
                                    CORP., a Delaware corporation


                                        By: /s/ Mark J. Warner                 
                                           ------------------------------------
                                                Mark J. Warner
                                                Agent and Attorney-in-Fact