<PAGE>   1
                                                                 EXHIBIT 10.34.1

                               GUARANTY AGREEMENT


                                       BY


                           BRIGHAM EXPLORATION COMPANY


                                   IN FAVOR OF


                           BANK OF MONTREAL, AS AGENT
                             AND EACH OF THE LENDERS
                          PARTY TO THE CREDIT AGREEMENT



                                JANUARY 26, 1998

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                                TABLE OF CONTENTS

<TABLE>
<CAPTION>

                                                                                                               Page
                                                                                                               ----
<S>                      <C>                                                                                    <C>
         ARTICLE 1

         General Terms

         Section 1.1       Terms Defined Above....................................................................1
         Section 1.2       Certain Definitions....................................................................1
         Section 1.3       Credit Agreement Definitions...........................................................2


         ARTICLE 2

         The Guaranty

         Section 2.1       Liabilities Guaranteed.................................................................2
         Section 2.2       Nature of Guaranty.....................................................................2
         Section 2.3       Agent's Rights.........................................................................3
         Section 2.4       Guarantor's Waivers....................................................................3
         Section 2.5       Maturity of Liabilities; Payment.......................................................4
         Section 2.6       Agent's Expenses.......................................................................4
         Section 2.7       Liability..............................................................................4
         Section 2.8       Events and Circumstances Not Reducing or Discharging Guarantor's
                           Obligations............................................................................4


         ARTICLE 3

         Representations, Warranties and Covenants


         Section 3.1       By Guarantor...........................................................................6
         Section 3.2       No Representation by Lenders..........................................................10

         ARTICLE 4

         Subordination of Indebtedness

         Section 4.1       Subordination of All Guarantor Claims.................................................10
         Section 4.2       Claims in Bankruptcy..................................................................10
         Section 4.3       Payments Held in Trust................................................................11
         Section 4.4       Liens Subordinate.....................................................................11
         Section 4.5       Guarantor's Enforcement Rights........................................................11
</TABLE>

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<TABLE>

<S>                      <C>                                                                                   <C>
         ARTICLE 5

         Certain Additional Covenants

         Section 5.1       Affirmative Covenants.................................................................11
         Financial Statements....................................................................................11
         Section 5.2       Negative Covenants....................................................................12


         ARTICLE 6

         Miscellaneous

         Section 6.1       Successors and Assigns................................................................17
         Section 6.2       Notices...............................................................................17
         Section 6.3       Business and Financial Information....................................................17
         Section 6.4       Construction..........................................................................17
         Section 6.5       Invalidity............................................................................17
         Section 6.6       ENTIRE AGREEMENT......................................................................17
</TABLE>


                                       ii


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                               GUARANTY AGREEMENT

     THIS GUARANTY AGREEMENT by BRIGHAM EXPLORATION COMPANY (hereinafter called
"Guarantor"), is in favor of BANK OF MONTREAL, as agent (the "Agent") for the
lenders (the "Lenders") that are or become parties to the Credit Agreement
defined below, and the Lenders.

                              W I T N E S S E T H:

     WHEREAS, on even date herewith, BRIGHAM OIL & GAS, L.P., a Delaware limited
partnership (hereinafter called "Borrower"), the Agent and the Lenders have
entered into that certain Credit Agreement (as the same may be amended from time
to time, the "Credit Agreement"); and

     WHEREAS, one of the terms and conditions stated in the Credit Agreement for
the making of the loans described therein is the execution and delivery to the
Agent for the benefit of the Lenders of this Guaranty Agreement;

     NOW, THEREFORE, (i) in order to comply with the terms and conditions of the
Credit Agreement, (ii) to induce the Lenders, at any time or from time to time,
to loan monies, with or without security to or for the account of Borrower in
accordance with the terms of the Credit Agreement, (iii) at the special
insistence and request of the Lenders, and (iv) for other good and valuable
consideration, the receipt and sufficiency of which is hereby acknowledged,
Guarantor hereby agrees as follows:

                                   ARTICLE I

                                 General Terms

     Section 1.1 Terms Defined Above. As used in this Guaranty Agreement, the
terms "Borrower", "Guarantor", "Credit Agreement" and "Lenders" shall have the
meanings indicated above.

     Section 1.2 Certain Definitions. As used in this Guaranty Agreement, the
following terms shall have the following meanings, unless the context otherwise
requires:

     "Guarantor Claims" shall have the meaning indicated in Section 4.1 hereof.

     "Guaranty Agreement" shall mean this Guaranty Agreement, as the same may
from time to time be amended or supplemented.

     "Liabilities" shall mean
<PAGE>   5
          (a) any and all indebtedness, obligations and liabilities now or
     hereafter incurred by the Borrower pursuant to the Credit Agreement,
     including without limitation, the unpaid fees and principal of and interest
     on the Notes;

          (b) payment of and performance of any and all present or future
     obligations of Borrower to a Lender according to the terms of any Hedging
     Agreement;

          (c) performance of all Letter of Credit Agreements executed from time
     to time by Borrower under or pursuant to the Credit Agreement and all
     reimbursement obligations for drawn or undrawn portions under any Letter of
     Credit now outstanding or hereafter issued under or pursuant to the Credit
     Agreement; and

          (d) any additional "Indebtedness", as defined in the Credit Agreement,

          (e) all renewals, rearrangements, increases, extensions for any
     period, amendments or supplement in whole or in part of the Notes or any
     documents evidencing the above.

          "Subsidiary Guarantor" shall mean a Subsidiary of the Guarantor which
     is itself a "Guarantor" under the Credit Agreement.

     Section 1.3 Credit Agreement Definitions. Unless otherwise defined herein,
all terms beginning with a capital letter which are defined in the Credit
Agreement shall have the same meanings herein as therein.

                                   ARTICLE 2

                                  The Guaranty

     Section 2.1 Liabilities Guaranteed. Guarantor hereby irrevocably and
unconditionally guarantees the prompt payment at maturity of the Liabilities.

     Section 2.2 Nature of Guaranty. This Guaranty Agreement is an absolute,
irrevocable, completed and continuing guaranty of payment and not a guaranty of
collection, and no notice of the Liabilities or any extension of credit already
or hereafter contracted by or extended to Borrower need be given to Guarantor.
This Guaranty Agreement may not be revoked by Guarantor and shall continue to be
effective with respect to debt under the Liabilities arising or created after
any attempted revocation by Guarantor and shall remain in full force and effect
until the Liabilities are paid in full and the Commitments are terminated,
notwithstanding that from time to time prior thereto no Liabilities may be
outstanding. Borrower and the Lenders may modify, alter, rearrange, extend for
any period and/or renew from time to time, the Liabilities, and the Lenders may
waive any Default or Events of Default without notice to the Guarantor and in
such event Guarantor will remain



                                       2
<PAGE>   6
fully bound hereunder on the Liabilities. This Guaranty Agreement shall continue
to be effective or be reinstated, as the case may be, if at any time any payment
of the Liabilities is rescinded or must otherwise be returned by any of the
Lenders upon the insolvency, bankruptcy or reorganization of Borrower or
otherwise, all as though such payment had not been made. This Guaranty Agreement
may be enforced by the Agent and any subsequent holder of any of the Liabilities
and shall not be discharged by the assignment or negotiation of all or part of
the Liabilities. Guarantor hereby expressly waives presentment, demand, notice
of non-payment, protest and notice of protest and dishonor, notice of Default or
Event of Default, and also notice of acceptance of this Guaranty Agreement,
acceptance on the part of the Lenders being conclusively presumed by the
Lenders' request for this Guaranty Agreement and delivery of the same to the
Agent.

     Section 2.3 Agent's Rights. Guarantor authorizes the Agent, without notice
or demand and without affecting Guarantor's liability hereunder, to take and
hold security for the payment of this Guaranty Agreement and/or the Liabilities,
and exchange, enforce, waive and release any such security; and to apply such
security and direct the order or manner of sale thereof as the Agent in its
discretion may determine; and to obtain a guaranty of the Liabilities from any
one or more Persons and at any time or times to enforce, waive, rearrange,
modify, limit or release any of such other Persons from their obligations under
such guaranties.

     Section 2.4 Guarantor's Waivers.

     (a) General. Guarantor waives any right to require any of the Lenders to
(i) proceed against Borrower or any other person liable on the Liabilities, (ii)
enforce any of their rights against any other guarantor of the Liabilities (iii)
proceed or enforce any of their rights against or exhaust any security given to
secure the Liabilities (iv) have Borrower joined with Guarantor in any suit
arising out of this Guaranty Agreement and/or the Liabilities, or (v) pursue any
other remedy in the Lenders' powers whatsoever. The Lenders shall not be
required to mitigate damages or take any action to reduce, collect or enforce
the Liabilities. Guarantor waives any defense arising by reason of any
disability, lack of corporate authority or power, or other defense of Borrower
or any other guarantor of the Liabilities, and shall remain liable hereon
regardless of whether Borrower or any other guarantor be found not liable
thereon for any reason. Whether and when to exercise any of the remedies of the
Lenders under any of the Loan Documents shall be in the sole and absolute
discretion of the Agent, and no delay by the Agent in enforcing any remedy,
including delay in conducting a foreclosure sale, shall be a defense to the
Guarantor's liability under this Guaranty Agreement.

     (b) Subrogation. Until the Liabilities have been paid in full, the
Guarantor waives all rights of subrogation or reimbursement against the
Borrower, whether arising by contract or operation of law (including, without
limitation, any such right arising under any federal or state bankruptcy or
insolvency laws) and waives any right to enforce any remedy which the Lenders
now have or may hereafter have against the Borrower, and waives any benefit or
any right to participate in any security now or hereafter held by the Agent or
any Lender.



                                       3
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     Section 2.5 Maturity of Liabilities; Payment. Guarantor agrees that if the
maturity of any of the Liabilities is accelerated by bankruptcy or otherwise,
such maturity shall also be deemed accelerated for the purpose of this Guaranty
Agreement without demand or notice to Guarantor. Guarantor will, forthwith upon
notice from the Agent, pay to the Agent the amount due and unpaid by Borrower
and guaranteed hereby. The failure of the Agent to give this notice shall not in
any way release Guarantor hereunder.

     Section 2.6 Agent's Expenses. If Guarantor fails to pay the Liabilities
after notice from the Agent of Borrower's failure to pay any Liabilities at
maturity, and if the Agent obtains the services of an attorney for collection of
amounts owing by Guarantor hereunder, or obtaining advice of counsel in respect
of any of their rights under this Guaranty Agreement, or if suit is filed to
enforce this Guaranty Agreement, or if proceedings are had in any bankruptcy,
probate, receivership or other judicial proceedings for the establishment or
collection of any amount owing by Guarantor hereunder, or if any amount owing by
Guarantor hereunder is collected through such proceedings, Guarantor agrees to
pay to the Agent the Agent's reasonable attorneys' fees.

     Section 2.7 Liability. It is expressly agreed that the liability of the
Guarantor for the payment of the Liabilities guaranteed hereby shall be primary
and not secondary.

     Section 2.8 Events and Circumstances Not Reducing or Discharging
Guarantor's Obligations. Guarantor hereby consents and agrees to each of the
following to the fullest extent permitted by law, and agrees that Guarantor's
obligations under this Guaranty Agreement shall not be released, diminished,
impaired, reduced or adversely affected by any of the following, and waives any
rights (including without limitation rights to notice) which Guarantor might
otherwise have as a result of or in connection with any of the following:

     (a) Modifications; etc. Any renewal, extension, modification, increase,
decrease, alteration or rearrangement of all or any part of the Liabilities, or
of the Notes, or the Credit Agreement or any instrument executed in connection
therewith, or any contract or understanding between Borrower and any of the
Lenders, or any other Person, pertaining to the Liabilities;

     (b) Adjustment, etc. Any adjustment, indulgence, forbearance or compromise
that might be granted or given by any of the Lenders to Borrower or Guarantor or
any Person liable on the Liabilities;

     (c) Condition of Borrower or Guarantor. The insolvency, bankruptcy
arrangement, adjustment, composition, liquidation, disability, dissolution,
death or lack of power of Borrower or Guarantor or any other Person at any time
liable for the payment of all or part of the Liabilities; or any dissolution of
Borrower or Guarantor, or any sale, lease or transfer of any or all of the
assets of Borrower or Guarantor, or any changes in the shareholders, partners,
or members of Borrower or Guarantor; or any reorganization of Borrower or
Guarantor;



                                       4
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     (d) Invalidity of Liabilities. The invalidity, illegality or 
unenforceability of all or any part of the Liabilities, or any document or
agreement executed in connection with the Liabilities, for any reason
whatsoever, including without limitation the fact that the Liabilities, or any
part thereof, exceed the amount permitted by law, the act of creating the
Liabilities or any part thereof is ultra vires, the officers or representatives
executing the documents or otherwise creating the Liabilities acted in excess of
their authority, the Liabilities violate applicable usury laws, the Borrower has
valid defenses, claims or offsets (whether at law, in equity or by agreement)
which render the Liabilities wholly or partially uncollectible from Borrower,
the creation, performance or repayment of the Liabilities (or the execution,
delivery and performance of any document or instrument representing part of the
Liabilities or executed in connection with the Liabilities, or given to secure
the repayment of the Liabilities) is illegal, uncollectible, legally impossible
or unenforceable, or the Credit Agreement or other documents or instruments
pertaining to the Liabilities have been forged or otherwise are irregular or not
genuine or authentic;

     (e) Release of Obligors. Any full or partial release of the liability of
Borrower on the Liabilities or any part thereof, of any co-guarantors, or any
other Person now or hereafter liable, whether directly or indirectly, jointly,
severally, or jointly and severally, to pay, perform, guarantee or assure the
payment of the Liabilities or any part thereof, it being recognized,
acknowledged and agreed by Guarantor that Guarantor may be required to pay the
Liabilities in full without assistance or support of any other Person, and
Guarantor has not been induced to enter into this Guaranty Agreement on the
basis of a contemplation, belief, understanding or agreement that other parties
other than the Borrower will be liable to perform the Liabilities, or the
Lenders will look to other parties to perform the Liabilities;

     (f) Other Security. The taking or accepting of any other security,
collateral or guaranty, or other assurance of payment, for all or any part of
the Liabilities;

     (g) Release of Collateral, etc. Any release, surrender, exchange,
subordination, deterioration, waste, loss or impairment (including without
limitation negligent, willful, unreasonable or unjustifiable impairment) of any
collateral, property or security, at any time existing in connection with, or
assuring or securing payment of, all or any part of the Liabilities;

     (h) Care and Diligence. The failure of the Lenders or any other Person to
exercise diligence or reasonable care in the preservation, protection,
enforcement, sale or other handling or treatment of all or any part of such
collateral, property or security;

     (i) Status of Liens. The fact that any collateral, security, security
interest or lien contemplated or intended to be given, created or granted as
security for the repayment of the Liabilities shall not be properly perfected or
created, or shall prove to be unenforceable or subordinate to any other security
interest or lien, it being recognized and agreed by Guarantor that Guarantor is
not entering into this Guaranty Agreement in reliance on, or in contemplation of
the benefits of, the validity, enforceability, collectibility or value of any of
the collateral for the Liabilities;




                                       5
<PAGE>   9
     (j) Payments Rescinded. Any payment by Borrower to the Lenders is held to
constitute a preference under the bankruptcy laws, or for any reason the Lenders
are required to refund such payment or pay such amount to Borrower or someone
else; or

     (k) Other Actions Taken or Omitted. Any other action taken or omitted to be
taken with respect to the Credit Agreement, the Liabilities, or the security and
collateral therefor, whether or not such action or omission prejudices Guarantor
or increases the likelihood that Guarantor will be required to pay the
Liabilities pursuant to the terms hereof; it being the unambiguous and
unequivocal intention of Guarantor that Guarantor shall be obligated to pay the
Liabilities when due, notwithstanding any occurrence, circumstance, event,
action, or omission whatsoever, whether contemplated or uncontemplated, and
whether or not otherwise or particularly described herein, except for the full
and final payment and satisfaction of the Liabilities.

                                   ARTICLE 3

                   Representations, Warranties and Covenants

     Section 3.1 By Guarantor. In order to induce the Lenders to accept this
Guaranty Agreement, Guarantor represents and warrants to the Lenders (which
representations and warranties will survive the creation of the Liabilities and
any extension of credit thereunder) that:

     (a) Benefit to Guarantor. Guarantor's guaranty pursuant to this Guaranty
Agreement reasonably may be expected to benefit, directly or indirectly,
Guarantor.

     (b) Corporate Existence. Guarantor is a corporation duly organized, legally
existing and in good standing under the laws of the State of Delaware and is
duly qualified in all jurisdictions wherein the property owned or the business
transacted by it makes such qualification necessary.

     (c) Corporate Power and Authorization. Guarantor is duly authorized and
empowered to execute, deliver and perform this Guaranty Agreement and all
corporate action on Guarantor's part requisite for the due execution, delivery
and performance of this Guaranty Agreement has been duly and effectively taken.

     (d) Binding Obligations. This Guaranty Agreement constitutes valid and
binding obligations of Guarantor, enforceable in accordance with its terms
(except that enforcement may be subject to any applicable bankruptcy, insolvency
or similar laws generally affecting the enforcement of creditors' rights) or by
general principles of equity.

     (e) No Consent. Guarantor's execution, delivery and performance of this
Guaranty Agreement does not require the consent or approval of any other Person,
including without limitation any regulatory authority or governmental body of
the United States or any state thereof or any political subdivision of the
United States or any state thereof.



                                       6
<PAGE>   10
     (f) Solvency. The Guarantor hereby represents that (i) it is not insolvent
as of the date hereof and will not be rendered insolvent as a result of this
Guaranty Agreement, (ii) it is not engaged in business or a transaction, or
about to engage in a business or a transaction, for which any property or assets
remaining with such Guarantor is unreasonably small capital, and (iii) it does
not intend to incur, or believe it will incur, debts that will be beyond its
ability to pay as such debts mature.

     (g) Financial Condition. The audited consolidated balance sheet of the
Guarantor and its Consolidated Subsidiaries as at December 31, 1996 and the
related consolidated statement of income, stockholders' equity and cash flow of
the Guarantor and its Consolidated Subsidiaries for the fiscal year ended on
said date, with the opinion thereon of Price Waterhouse heretofore furnished to
each of the Lenders and the unaudited consolidated balance sheet of the
Guarantor and its Consolidated Subsidiaries as at September 30, 1997 and their
related consolidated statements of income, stockholders' equity and cash flow of
the Guarantor and its Consolidated Subsidiaries for the nine-month period ended
on such date heretofore furnished to the Agent, are complete and correct and
fairly present the consolidated financial condition of the Guarantor and its
Consolidated Subsidiaries as at said dates and the results of its operations for
the fiscal year and the nine-month period on said dates, all in accordance with
GAAP, as applied on a consistent basis (subject, in the case of the interim
financial statements, to normal year-end adjustments). Neither the Guarantor nor
any of its Subsidiaries has on the Closing Date any material Debt, Trade
Payables, contingent liabilities, liabilities for taxes, unusual forward or
long-term commitments or unrealized or anticipated losses from any unfavorable
commitments, except as referred to or reflected or provided for in the Financial
Statements or in Schedule 7.02 of the Credit Agreement. Since December 31,
1996, there has been no change or event having a Material Adverse Effect. Since
the date of the Financial Statements, neither the business nor the Properties of
the Guarantor and its Consolidated Subsidiaries, taken as a whole, have been
materially and adversely affected as a result of any fire, explosion,
earthquake, flood, drought, windstorm, accident, strike or other labor
disturbance, embargo, requisition or taking of Property or cancellation of
contracts, permits or concessions by any Governmental Authority, riot,
activities of armed forces or acts of God or of any public enemy.

     (h) Litigation. Except as disclosed to the Lenders in Schedule 7.03 of the
Credit Agreement, at the Closing Date there is no litigation, legal,
administrative or arbitral proceeding, investigation or other action of any
nature pending or, to the knowledge of the Guarantor threatened against or
affecting the Guarantor or any of its Subsidiaries which both (a) involves the
possibility of any judgment or liability against the Guarantor or any of its
Subsidiaries not fully covered by insurance (except for normal deductibles), and
(b) would be more likely than not to have a Material Adverse Effect.

     (i) No Breach. Neither the execution and delivery of this Guaranty
Agreement, nor compliance with the terms and provisions hereof will conflict
with or result in a breach of, or require any consent which has not been
obtained as of the Closing Date under, the respective charter or by-laws of the
Guarantor or any of its Subsidiaries or any Governmental Requirement or any
material agreement or instrument to which the Guarantor or any of its
Subsidiaries is a party or by which it



                                       7
<PAGE>   11
is bound or to which it or its Properties are subject, or constitute a default
under any such agreement or instrument, or result in the creation or imposition
of any Lien upon any of the material revenues or assets of the Guarantor or any
of its Subsidiaries pursuant to the terms of any such agreement or instrument
other than the Liens created by the Loan Documents.

     (j) ERISA.

          (1) The Guarantor, its Subsidiaries and each ERISA Affiliate have
     complied in all material respects with ERISA and, where applicable, the
     Code regarding each Plan.

          (2) Each Plan is, and has been, maintained in substantial compliance
     with ERISA and, where applicable, the Code.

          (3) No act, omission or transaction has occurred which could result in
     imposition on the Guarantor, any of its Subsidiaries or any ERISA Affiliate
     (whether directly or indirectly) of an amount of $100,000 or more as (i)
     either a civil penalty assessed pursuant to section 502(c), (i) or (l) of
     ERISA or a tax imposed pursuant to Chapter 43 of Subtitle D of the Code or
     (ii) breach of fiduciary duty liability damages under section 409 of ERISA.

          (4) No Plan (other than a defined contribution plan) or any trust
     created under any such Plan has been terminated since September 2, 1974. No
     liability to the PBGC in excess of $100,000 (other than for the payment of
     current premiums which are not past due) by the Guarantor, any of its
     Subsidiaries or any ERISA Affiliate has been or is expected by the
     Guarantor, any of its Subsidiaries or any ERISA Affiliate to be incurred
     with respect to any Plan. No ERISA Event with respect to any Plan has
     occurred which could reasonably expected to result in liabilities of 
     $100,000 or more.

          (5) Full payment when due has been made of all amounts which the
     Guarantor, any of its Subsidiaries or any ERISA Affiliate is required under
     the terms of each Plan or applicable law to have paid as contributions to
     such Plan, and no accumulated funding deficiency in an amount of $100,000
     or more (as defined in section 302 of ERISA and section 412 of the Code),
     whether or not waived, exists with respect to any Plan.

          (6) The actuarial present value of the benefit liabilities under each
     Plan which is subject to Title IV of ERISA does not, as of the end of the
     Guarantor's most recently ended fiscal year, exceed the current value of
     the assets (computed on a plan termination basis in accordance with Title
     IV of ERISA) of such Plan allocable to such benefit liabilities by $100,000
     or more. The term "actuarial present value of the benefit liabilities"
     shall have the meaning specified in section 4041 of ERISA.



                                       8
<PAGE>   12
          (7) None of the Guarantor, any of its Subsidiaries or any ERISA
     Affiliate sponsors, maintains, or contributes to an employee welfare
     benefit plan, as defined in section 3(l) of ERISA, including, without
     limitation, any such plan maintained to provide benefits to former
     employees of such entities, that may not be terminated by the Guarantor,
     any of its Subsidiaries or any ERISA Affiliate in its sole discretion at
     any time without any material liability.

          (8) None of the Guarantor, any of its Subsidiaries or any ERISA
     Affiliate sponsors, maintains or contributes to, or has at any time in the
     preceding six calendar years, sponsored, maintained or contributed to, any
     Multiemployer Plan.

          (9) None of the Guarantor, any of its Subsidiaries or any ERISA
     Affiliate is required to provide security under section 401(a)(29) of the
     Code due to a Plan amendment that results in an increase in current
     liability for the Plan.

     (l) Taxes. Except as set out in Schedule 7.09 of the Credit Agreement, each
of the Guarantor and its Subsidiaries has filed all United States Federal income
tax returns and all other tax returns which are required to be filed by it and
has paid all material taxes due pursuant to such returns or pursuant to any
assessment received by the Guarantor or any of its Subsidiaries, except for any
taxes which are being contested in good faith and by proper proceedings and
against which adequate reserves are being maintained. The charges, accruals and
reserves on the books of the Guarantor and its Subsidiaries in respect of taxes
and other governmental charges are, in the opinion of the Guarantor, adequate.
No tax lien has been filed and, to the knowledge of the Guarantor, no claim is
being asserted with respect to any such tax, fee or other charge, except for any
taxes, fees or other charges which are being contested in good faith and by
proper proceedings and against which adequate reserves are being maintained.

     (m) Title to Ownership Interest in General Partner and Limited Partners of
Borrower. The Guarantor has good and unencumbered title to all of the capital
stock of the General Partner of the Borrower and, directly or indirectly, all of
the ownership interests in the limited partners of the Borrower.

     (n) Investment Company Act. Neither the Guarantor nor any of its
Subsidiaries is an "investment company" or a company "controlled" by an
"investment company," within the meaning of the Investment Company Act of 1940,
as amended.

     (o) Public Utility Holding Company Act. Neither the Guarantor nor any of
its Subsidiaries is a "holding company," or a "subsidiary company" of a "holding
company," or an "affiliate" of a "holding company" or of a "subsidiary company"
of a "holding company," or a "public utility" within the meaning of the Public
Utility Holding Company Act of 1935, as amended.

     (p) Subsidiaries. Except as set forth on Schedule I hereto or as permitted
by Section 9.16 of the Credit Agreement, the Guarantor has no Subsidiaries.




                                       9
<PAGE>   13
     (q) Compliance with the Law. Neither the Guarantor nor any of its
Subsidiaries has violated any Governmental Requirement or failed to obtain any
license, permit, franchise or other governmental authorization necessary for the
ownership of any of its Properties or the conduct of its business, which
violation or failure would have (in the event such violation or failure were
asserted by any Person through appropriate action) a Material Adverse Effect.

     Section 3.2 No Representation by Lenders. Neither the Lenders nor any other
Person has made any representation, warranty or statement to the Guarantor in
order to induce the Guarantor to execute this Guaranty Agreement.

                                   ARTICLE 4

                         Subordination of Indebtedness

     Section 4.1 Subordination of All Guarantor Claims. As used herein, the term
"Guarantor Claims" shall mean all debts and liabilities of Borrower or any
Subsidiary of the Borrower to Guarantor, whether such debts and liabilities now
exist or are hereafter incurred or arise, or whether the obligation of Borrower
or such Subsidiary thereon be direct, contingent, primary, secondary, several,
joint and several, or otherwise, and irrespective of whether such debts or
liabilities be evidenced by note, contract, open account, or otherwise, and
irrespective of the person or persons in whose favor such debts or liabilities
may, at their inception, have been, or may hereafter be created, or the manner
in which they have been or may hereafter be acquired by Guarantor. The Guarantor
Claims shall include without limitation all rights and claims of Guarantor
against Borrower or any Subsidiary of the Borrower arising as a result of
subrogation or otherwise as a result of Guarantor's payment of all or a portion
of the Liabilities. Until the Liabilities shall be paid and satisfied in full
and Guarantor shall have performed all of its obligations hereunder, Guarantor
shall not receive or collect, directly or indirectly, from Borrower or any
Subsidiary of the Borrower or any other party any amount upon the Guarantor
Claims.

     Section 4.2 Claims in Bankruptcy. In the event of receivership, bankruptcy,
reorganization, arrangement, debtor's relief, or other insolvency proceedings
involving Borrower or any Subsidiary of the Borrower, as debtor, the Lenders
shall have the right to prove their claim in any proceeding, so as to establish
their rights hereunder and receive directly from the receiver, trustee or other
court custodian, dividends and payments which would otherwise be payable upon
Guarantor Claims. Guarantor hereby assigns such dividends and payments to the
Lenders. Should the Agent or any Lender receive, for application upon the
Liabilities, any such dividend or payment which is otherwise payable to
Guarantor, and which, as between Borrower or any Subsidiary of the Borrower and
Guarantor, shall constitute a credit upon the Guarantor Claims, then upon
payment in full of the Liabilities, Guarantor shall become subrogated to the
rights of the Lenders to the extent that such payments to the Lenders on the
Guarantor Claims have contributed toward the liquidation of the Liabilities, and
such subrogation shall be with respect to that proportion of the Liabilities
which




                                       10
<PAGE>   14
would have been unpaid if the Agent or a Lender had not received dividends or
payments upon the Guarantor Claims.

     Section 4.3 Payments Held in Trust. In the event that notwithstanding
Sections 4.1 and 4.2 above, Guarantor should receive any funds payments, claims
or distributions which is prohibited by such Sections, Guarantor agrees to hold
in trust for the Lenders an amount equal to the amount of all funds, payments,
claims or distributions so received, and agrees that it shall have absolutely no
dominion over the amount of such funds, payments, claims or distributions except
to pay them promptly to the Agent, and Guarantor covenants promptly to pay the
same to the Agent.

     Section 4.4 Liens Subordinate. Guarantor agrees that any liens, security
interests, judgment liens, charges or other encumbrances upon Borrower's or any
Subsidiary of the Borrower's assets securing payment of the Guarantor Claims
shall be and remain inferior and subordinate to any liens, security interests,
judgment liens, charges or other encumbrances upon Borrower's or any Subsidiary
of the Borrower's assets securing payment of the Liabilities, regardless of
whether such encumbrances in favor of Guarantor, the Agent or the Lenders
presently exist or are hereafter created or attach.

     Section 4.5 Guarantor's Enforcement Rights. Without the prior written
consent of the Lenders, Guarantor shall not (a) exercise or enforce any
creditor's right it may have against the Borrower or any Subsidiary of the
Borrower, or (b) foreclose, repossess, sequester or otherwise take steps or
institute any action or proceeding (judicial or otherwise, including without
limitation the commencement of or joinder in any liquidation, bankruptcy,
rearrangement, debtor's relief or insolvency proceeding) to enforce any lien,
mortgages, deeds of trust, security interest, collateral rights, judgments or
other encumbrances on assets of Borrower or any Subsidiary of the Borrower held
by Guarantor.

                                   ARTICLE 5

                          Certain Additional Covenants

     Section 5.1 Affirmative Covenants.

          Financial Statements. The Guarantor shall cause the Borrower to
deliver as and when required, the financial statements and other information
called for in Section 8.01 of the Credit Agreement and Guarantor shall:

               (i) Notice of Default. Promptly after the Guarantor knows that
          any default in any covenant herein has occurred, the Guarantor will
          give notice of such default to the Agent, describing the same in
          reasonable detail and the action the Guarantor proposes to take with
          respect thereto.


                                       11

<PAGE>   15
               (ii) Payment of Trade Payables. The Guarantor will pay and cause
          all of its Subsidiaries to pay all of their respective Trade Payables
          now or hereafter incurred within 60 days of the invoice or billing
          date, unless subject to legal offset or unless being contested in good
          faith by appropriate proceedings and reserves adequate under GAAP
          shall have been established therefor.

     Section 5.2 Negative Covenants. The Guarantor covenants and agrees that, so
long as any of the Commitments are in effect and until payment in full of the
Liabilities hereunder, all interest thereon and all other amounts payable by the
Guarantor hereunder, without the prior written consent of the Majority Lenders:

     (a) Debt. The Guarantor will not and will not permit any of its
Subsidiaries to incur, create, assume or suffer to exist any Debt, except:

               (1) the Notes or other Indebtedness arising under the Loan
          Documents or any guaranty of or suretyship arrangement for the Notes
          or other Indebtedness arising under the Loan Documents;

               (2) Debt existing on the Closing Date which is reflected in the
          Financial Statements or is disclosed in Schedule 9.01 to the Credit
          Agreement, and any renewals or extensions (but not increases) thereof;

               (3) Debt permitted under the Credit Agreement;

               (4) Debt owing to the Borrower, the Guarantor or a Subsidiary
          Guarantor which (other than Debt owing to the Borrower) is
          subordinated to the Indebtedness as provided in the Guaranty Agreement
          of the Guarantor or such Subsidiary Guarantor; and

               (5) Debt of the Guarantor not described in Sections 5.2 (1)
          through (4) above which together with all Debt of the Borrower allowed
          under Section 9.01(h) of the Credit Agreement does not exceed
          $1,000,000 at any one time outstanding.

     (b) Liens. The Guarantor will not and will not permit any of its
Subsidiaries to create, incur, assume or permit to exist any Lien on any of its
Properties (now owned or hereafter acquired), except:

               (1) Liens securing the payment of any Indebtedness;

               (2) Excepted Liens;

               (3) Liens incurred by the Borrower and the Borrower's
          Subsidiaries as permitted under the Credit Agreement.


                                       12

<PAGE>   16
     (c) Investments, Loans and Advances. The Guarantor will not and will not
permit any of its Subsidiaries to remain outstanding any loans or advances to or
investments in any Person, except that the foregoing restriction shall not apply
to:

          (1)  investments, loans or advances by the Guarantor or a Subsidiary
               Guarantor (other than a Subsidiary of the Borrower) in or to the
               Borrower, the Guarantor or a Subsidiary Guarantor; and

          (2)  any other investments, loans and advances by the Borrower or a
               Subsidiary of the Borrower as permitted under Section 9.03 of the
               Credit Agreement.

     (d) Dividends, Distributions and Redemptions. The Guarantor will not
declare or pay any dividend, purchase, redeem or otherwise acquire for value any
of its stock now or hereafter outstanding, return any capital to its
stockholders or make any distribution of its assets to its stockholders.

     (e) Sales and Leasebacks. The Guarantor will not and will not permit any of
its Subsidiaries to enter into any arrangement, directly or indirectly, with any
Person whereby such Person shall sell or transfer any of its Property, whether
now owned or hereafter acquired, and whereby such Person shall then or
thereafter rent or lease as lessee such Property or any part thereof or other
Property which such Person intends to use for substantially the same purpose or
purposes as the Property sold or transferred.

     (f) Nature of Business. The Guarantor will not allow any material change to
be made in the character of its business as presently conducted.

     (g) Limitation on Leases. The Guarantor will not and will not permit any of
its Subsidiaries to create, incur, assume or suffer to exist any obligation for
the payment of rent or hire of Property of any kind whatsoever (real or personal
including capital leases but excluding leases of Hydrocarbon Interests and the
equipment used thereon), under leases or lease agreements which would cause the
aggregate amount of all payments made by the Guarantor and its Subsidiaries
pursuant to all such leases or lease agreements to exceed $2,000,000 in any
period of twelve consecutive calendar months during the life of such leases.

     (h) Mergers, Etc. The Guarantor will not and will not permit any of its
Subsidiaries to merge into or with or consolidate with any other Person or sell,
lease or otherwise dispose of (whether in one transaction or in a series of
transactions) all or substantially all of its Property or assets to any other
Person whose the Guarantor, the Borrower or any Subsidiary Guarantor unless
Guaranty Agreement is in full force and effect is the surviving entity and no
Default exists or will be created thereby.



                                       13
<PAGE>   17
     (i) ERISA Compliance. The Guarantor will not and will not permit any of its
Subsidiaries to at any time:

          (1) Engage in, or permit any ERISA Affiliate to engage in, any
     transaction in connection with which the Guarantor, any of its Subsidiaries
     or any ERISA Affiliate could be subjected to either a civil penalty
     assessed pursuant to section 502(c), (i) or (1) of ERISA or a tax imposed
     by Chapter 43 of Subtitle D of the Code in excess of $100,000;

          (2) Terminate, or permit any ERISA Affiliate to terminate, any Plan in
     a manner, or take any other action with respect to any Plan, which could
     reasonably be expected to result in any liability of the Guarantor, any of
     its Subsidiaries or any ERISA Affiliate to the PBGC in excess of $ 100,000;

          (3) Fail to make, or permit any ERISA Affiliate to fail to make, full
     payment when due of all amounts which under the provisions of any Plan,
     agreement relating thereto or applicable law, the Guarantor, any of its
     Subsidiaries or any ERISA Affiliate is required to pay as contributions
     thereto;

          (4) Permit to exist, or allow any ERISA Affiliate to permit to exist,
     any accumulated funding deficiency in excess of $100,000 within the
     meaning of Section 302 of ERISA or section 412 of the Code, whether or not
     waived, with respect to any Plan;

          (5) Permit, or allow any ERISA Affiliate to permit, the actuarial
     present value of the benefit liabilities under any Plan maintained by the
     Guarantor, any of its Subsidiaries or any ERISA Affiliate which is
     regulated under Title IV of ERISA to exceed the current value of the assets
     (computed on a plan termination basis in accordance with Title IV of ERISA)
     of such Plan allocable to such benefit liabilities by an amount in excess
     of $100,000. The term "actuarial present value of the benefit liabilities"
     shall have the meaning specified in section 4041 of ERISA;

          (6) Contribute to or assume an obligation to contribute to, or permit
     any ERISA Affiliate to contribute to or assume an obligation to contribute
     to, any Multiemployer Plan;

          (7) Acquire, or permit any ERISA Affiliate to acquire, an interest in
     any Person that causes such Person to become an ERISA Affiliate with
     respect to the Borrower or any ERISA Affiliate if such Person sponsors,
     maintains or contributes to, or at any time in the six-year period
     preceding such acquisition has sponsored, maintained, or contributed to,
     (a) any Multiemployer Plan, or (b) any other Plan that is subject to Title
     IV of ERISA under which the actuarial present value of the benefit
     liabilities under such Plan exceeds the current value of the assets
     (computed on a plan termination


                                       14

<PAGE>   18
     basis in accordance with Title IV of ERISA) of such Plan allocable to such
     benefit liabilities;

          (8) Incur, or permit any ERISA Affiliate to incur, a liability to or
     on account of a Plan under sections 515, 4062, 4063, 4064, 4201 or 4204 of
     ERISA which in the aggregate for all such liability exceeds $100,000;

          (9) Contribute to or assume an obligation to contribute to, or permit
     any ERISA Affiliate to contribute to or assume an obligation to contribute
     to, any employee welfare benefit plan, as defined in section 3(l) of ERISA,
     including, without limitation, any such plan maintained to provide benefits
     to former employees of such entities, that may not be terminated by such
     entities in their sole discretion at any time without any material
     liability; or

          (10) Amend or permit any ERISA Affiliate to amend, a Plan resulting in
     an increase in current liability such that the Guarantor, any of its
     Subsidiaries or any ERISA Affiliate is required to provide security to such
     Plan under section 401(a)(29) of the Code.

     (j) Sale or Discount of Receivables. The Guarantor will not and will not
permit any of its Subsidiaries to discount or sell (with or without recourse)
any of its notes receivable or accounts receivable.

     (k) Capital Expenditures. The Guarantor will not and will not permit any of
its Subsidiaries to make any expenditures to acquire non cash flow generating
fixed or capital assets such as seismic data and unproved Oil and Gas Properties
if, after giving effect thereto, the aggregate of all such expenditures would
exceed $21,000,000 from January 1, 1998, until the date the Initial Borrowing
Base has been set. Costs of drilling and development activities will not be
considered "acquisition expenditures" for the purposes of this section.

     (1) Environmental Matters. The Guarantor will not and will not permit any
of its Subsidiaries to cause or permit any of its Property to be in violation
of, or do anything or permit anything to be done which will subject any such
Property to any remedial obligations under any Environmental Laws, assuming
disclosure to the applicable Governmental Authority of all relevant facts,
conditions and circumstances, if any, pertaining to such Property where such
violations or remedial obligations would have a Material Adverse Effect.

     (m) Transactions with Affiliates. The Guarantor will not and will not
permit any of its Subsidiaries to enter into any transaction, including, without
limitation, any purchase, sale, lease or exchange of Property or the rendering
of any service, with any Affiliate unless such transactions are otherwise not in
violation of the Credit Agreement, are in the ordinary course of its business
and are upon fair and reasonable terms no less favorable to it (or, if the
Borrower is involved, no less


                                       15

<PAGE>   19
favorable to the Borrower) than it would obtain in a comparable arm's length
transaction with a Person not an Affiliate.

     (n) Subsidiaries. The Guarantor will not and will not permit any of its
Subsidiaries to, create any additional Subsidiaries except for the Subsidiaries
of the Borrower as permitted by the Credit Agreement.

     (o) Negative Pledge Agreements. The Guarantor will not and will not permit
any of its Subsidiaries to create, incur, assume or suffer to exist any
contract, agreement or understanding (other than the Loan Documents) which in
any way prohibits or restricts the granting, conveying, creation or imposition
of any Lien on any of its Property or which requires the consent of or notice to
other Persons in connection therewith.

     (p) Capital Stock of General Partner of Borrower. The Guarantor will not
permit any of the capital stock of the General Partner of the Borrower or any
ownership interest in any limited partner of the Borrower to be owned or
controlled by any Person other than the Guarantor or Brigham Inc..

     (q) Current Ratio. The Guarantor will not permit its ratio of (i)
consolidated current assets of the Guarantor and its Consolidated Subsidiaries
(including, without limitation, any unused and available Aggregate Commitments)
to (ii) their consolidated current liabilities (excluding the Indebtedness) to
be less than 1.0 to 1.0 at any time.

     (r) Tangible Net Worth. The Guarantor will not permit its Tangible Net 
Worth to be less than $35,000,000 plus 75% of the net proceeds of new equity
offerings plus 75% of Consolidated Net Income, if positive for the fourth
quarter 1997 and for any fiscal year of the Guarantor beginning with fiscal year
1998. This Tangible Net Worth covenant will be tested at the aforesaid dates and
at the end of each of the first three fiscal quarters of the Guarantor and
annually. For each quarterly test the interim net losses may be netted against
interim net income for the same reporting year to determine the net income, if
positive, to be used in such test for such reporting year.

     (s) Interest Coverage Ratio. The Guarantor will not permit its Interest
Coverage Ratio as of the end of any fiscal quarter of the Guarantor (calculated
quarterly at the end of each fiscal quarter) to be less than 3 to 1.00. Interest
Coverage Ratio shall mean the following ratios of (i) EBITDA to (ii) cash
interest payments accruing during the following periods:

          (i)   not less than 2 1/2 to 1 for the three (3) month period ending
                March 31, 1998;

          (ii)  not less than 2 1/2 to 1 for the six (6) month period ending
                June 30, 1998; and

          (iii) not less than 3 to 1 for the nine (9) month period ending
                September 30, 1998; and



                                       16
<PAGE>   20
          (iv) thereafter, not less than 3 to 1 for the twelve (12) month
               periods ending at the end of each fiscal quarter of the
               Guarantor.

                                   ARTICLE 6

                                 Miscellaneous

     Section 6.1 Successors and Assigns. This Guaranty Agreement is and shall
be, in every particular, available to the successors and assigns of the Lenders
and is and shall always be fully binding upon the legal representatives, heirs,
successors and assigns of Guarantor, notwithstanding that some or all of the
monies, the repayment of which this Guaranty Agreement applies, may be actually
advanced after any bankruptcy, receivership, reorganization, death, disability
or other event affecting Guarantor.

     Section 6.2 Notices. Any notice or demand to Guarantor under or in
connection with this Guaranty Agreement may be given and shall conclusively be
deemed and considered to have been given and received in accordance with Section
12.02 of the Credit Agreement, addressed to Guarantor at the address on the
signature page hereof or at such other address provided to the Agent in writing.

     Section 6.3 Business and Financial Information. Subject to any applicable
confidentiality agreements, the Guarantor will promptly furnish to the Agent and
the Lenders from time to time upon request such information regarding the
business and affairs and financial condition of the Guarantor and its
subsidiaries as the Agent and the Lenders may reasonably request.

     Section 6.4 Construction. This Guaranty Agreement is a contract made under
and shall be construed in accordance with and governed by the laws of the State
of Texas.

     Section 6.5 Invalidity. In the event that any one or more of the provisions
contained in this Guaranty Agreement shall, for any reason, be held invalid,
illegal or unenforceable in any respect, such invalidity, illegality or
unenforceability shall not affect any other provision of this Guaranty
Agreement.

     Section 6.6 ENTIRE AGREEMENT. THIS WRITTEN GUARANTY AGREEMENT EMBODIES THE
ENTIRE AGREEMENT AND UNDERSTANDING BETWEEN THE LENDERS AND THE GUARANTOR AND
SUPERSEDES ALL OTHER AGREEMENTS AND UNDERSTANDINGS BETWEEN SUCH PARTIES RELATING
TO THE SUBJECT MATTER HEREOF AND THEREOF. THIS WRITTEN GUARANTY AGREEMENT
REPRESENTS 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 AGREEMENTS BETWEEN THE PARTIES.



                                       17
<PAGE>   21
     WITNESS THE EXECUTION HEREOF, as of this the 26th day of January, 1998.



                                             BRIGHAM EXPLORATION COMPANY      
                                                                              
                                             By: /s/ Craig M. Fleming         
                                                ------------------------------
                                             Name: Craig M. Fleming           
                                             Title: Chief Financial Officer   
                                                                              
                                             Address for Notices:             
                                                                              
                                             6300 Bridge Point Parkway        
                                             Building 2, Suite 500            
                                             Austin, Texas 78730              
                                                                              
                                             Telecopier No.: (512) 427-3400   
                                             Telephone No.: (512) 427-3300    
                                             Attention: Craig M. Fleming      
                                             




                                       18
<PAGE>   22
                                   SCHEDULE I

                      SUBSIDIARIES OF BRIGHAM EXPLORATION

Brigham Exploration has the following Subsidiaries:

1.   Brigham, Inc., a Nevada corporation, all of whose shares are owned by
     Brigham Exploration.

2.   Brigham Holdings I, LLC, a Nevada limited liability company, whose only
     member is Brigham Exploration.

3.   Brigham Holdings II, LLC, a Nevada limited liability company, whose only
     member is Brigham, Inc.

4.   Borrower, a Delaware limited partnership, whose only General Partner is
     Brigham, Inc., and whose only limited partners are Brigham Holdings I, LLC
     and Brigham Holdings II, LLC.