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                                                                   EXHIBIT 10.8




                         REGISTRATION RIGHTS AGREEMENT


         This REGISTRATION RIGHTS AGREEMENT (the "Agreement") is made and
entered into as of February 17, 2000, by and between Brigham Exploration
Company, a Delaware corporation (the "Company"), and Shell Capital Inc., a
Delaware corporation (the "Purchaser").

                                    RECITALS

         WHEREAS, Brigham Oil & Gas, L.P., a Delaware limited partnership
("BOG") and the Purchaser are parties to the Amended and Restated Credit
Agreement of even date herewith (as modified from time to time, the "Credit
Agreement"), governing certain Notes as defined therein (the "Notes"), and such
Notes were issued to the Purchaser.

         WHEREAS, the Company has agreed to issue certain capital stock of the
Company ("Common Stock") in connection with the exercise of certain equity
conversion rights pursuant to that certain Equity Conversion Agreement of even
date herewith among the Company, the Purchaser, and BOG (the "Equity Conversion
Agreement") and in connection with Warrants issued pursuant to that certain
Warrant Agreement of even date herewith between the Company and the Purchaser
(the "Warrants").

         WHEREAS, the Purchaser desires that the Company register the Common
Stock issuable upon conversion of the Notes pursuant to the Equity Conversion
Agreement (the "Conversion Shares") and the Common Stock issuable upon the
exercise of the Warrants (the "Warrant Shares") upon the terms and subject to
the conditions set forth in this Agreement.

         WHEREAS, the execution and delivery of this Agreement is a condition
precedent to the Purchaser's obligations under the Credit Agreement.

         NOW, THEREFORE, the parties hereto, intending legally to be bound,
hereby agree as follows:

1.       DEFINITIONS.

         As used in this Agreement, the following terms shall have the meanings
ascribed to them below:

         "Applicable Rate" shall mean the Base Rate under the Credit Agreement
plus 6% per annum; provided that if the Credit Agreement shall have terminated
then the Base Rate shall be calculated for purposes of this Agreement as if
such Credit Agreement shall not have terminated.

         "BOG" shall have the meaning indicated above.



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         "Common Stock" shall have the meaning indicated above.

         "Company" shall have the meaning indicated above.

         "Conversion Shares" shall have the meaning indicated above.

         "Credit Agreement" shall have the meaning indicated above.

         "Equity Conversion Agreement" shall have the meaning indicated above.

         "Holder" shall mean the Purchaser, and any other Person who has become
a Permitted Transferee pursuant to Section 9(c).

         "Notes" shall have the meaning indicated above.

         "Purchaser" shall have the meaning indicated above.

         "Registrable Securities" means (a) the Conversion Shares, (b) the
Warrant Shares, (c) all other shares of Common Stock owned by the Purchaser and
its Affiliates as of the date of filing of the Registration Statement (or which
they shall have the right to purchase or acquire as of such date), and (d) any
securities issued or issuable in respect of or in exchange for any of the
shares of Common Stock referred to in clauses (a) through (c) above by way of a
stock dividend or stock split or in connection with a combination of shares,
recapitalization, reclassification, merger, consolidation, or exchange offer
("Distribution Securities"). For purposes of this Agreement, a Registrable
Security ceases to constitute a Registrable Security hereunder (i) when such
Registrable Security shall have been effectively registered under the
Securities Act and disposed of in a public market transaction pursuant to a
Registration Statement, (ii) when such Registrable Security shall have been
sold pursuant to Rule 144 (or any successor provision) under the Securities
Act, (iii) when such Registrable Security shall have been otherwise transferred
and a new certificate for such Registrable Security not bearing a legend
restricting further transfer shall have been delivered by the Company, (iv)
with respect to a particular Holder, at any time when all of such Holder's
remaining Registrable Securities can be sold in a single transaction in
compliance with Rule 144 under the Securities Act, (v) on the third anniversary
of the original issuance date of such Registrable Security, provided that the
Holder of such Registrable Security is not an Affiliate of the Company as of
such date, and has not been an Affiliate of the Company for a period of three
months preceding such date, or (vi) when such Registrable Security shall have
ceased to be outstanding.

         "Registration Statement" shall have the meaning set forth in Section
2.

         "Registration Termination Date" means, the first date on which all
Conversion Shares and Warrant Shares (and any Distribution Securities with
respect thereto) cease to be Registrable Securities.

         "Securities Act" means the Securities Act of 1933, as amended.

         "Securities Exchange Act" means the Securities Exchange Act of 1934,
         as amended.


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         "SEC" means the United States Securities Exchange Commission, or any
successor agency thereto.

         "Warrant Shares" shall have the meaning indicated above.

         "Warrants" shall have the meaning indicated above.

         Other capitalized terms shall have the meanings ascribed to them in
the other sections of this Agreement or in the Credit Agreement or in the
Equity Conversion Agreement in each case if not defined by the prior agreement
listed.

2.       REGISTRATION UPON CONVERSION.

         (a) Effective Registration. On or before sixty (60) days after the
date that any of the Conversion Shares or Warrant Shares are issued (the
"Filing Deadline"), the Company shall file with the SEC under the Securities
Act a Registration Statement on Form S-3 ("Registration Statement"), or on such
other registration form under the Securities Act as the Company shall deem
appropriate, covering the offer and sale by the Holders on a continuous or
delayed basis pursuant to Rule 415 thereunder (or any similar rule that may be
adopted by the SEC) of the Registrable Securities. The Company shall use its
best efforts to cause each Registration Statement to be declared effective on
or prior to the 120th day following the Filing Deadline and shall thereafter
keep such Registration Statement continuously effective until the Registration
Termination Date.

         (b) Delay in Effectiveness. In the event any Registration Statement is
not declared effective by the SEC within 120 days after the Filing Deadline
(the "Effectiveness Deadline"), or in the event the effectiveness of any
Registration Statement is suspended or terminated at any time after its
Effectiveness Deadline and prior to the Registration Termination Date, in each
case due to any action or inaction of the Company in pursuing, failing to
pursue, securing and maintaining an effective registration, utilizing all
reasonable efforts and good faith, or otherwise primarily for any reason
reasonably within the Company's control, then, at the election of the Holder:
(1) for each month (or portion thereof) such Registration Statement is not so
effective, the Company shall pay to the Holder, as liquidated damages and not
as a penalty, an amount equal to the product of (i) the Applicable Rate and
(ii) the greater of (x) the number of shares of Common Stock that are or could
become, if all Notes were converted and Warrants were exercised, Registrable
Securities multiplied by the closing price of the Common Stock on the first day
of the month as to which liquidated damages are payable as provided below, and
(y) outstanding principal amount of the Notes eligible for conversion to Common
Stock; or (2) the Holder may seek remedies at law or in equity. If there is
more than one Holder, the Company shall pay any damages to each Holder based
upon the proportion of the Registrable Securities owned by such Holder. Such
payments shall be made on the first Business Day of each month following any
month in which such Registration Statement is not effective, with a final
payment within five (5) Business Days after such Registration Statement becomes
effective.

3.       REGISTRATION PROCEDURES.

         (a) Company Procedures. In connection with the Company's registration
obligations pursuant to Section 2, the Company shall keep each Registration
Statement continuously effective for the period of time provided in Section 2,
to permit the sale of Registrable Securities covered by



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such Registration Statement in accordance with the intended method or methods
of distribution thereof specified in such Registration Statement or in the
related prospectus(es), and shall:

                         (i) comply with such provisions of the Securities Act
              as may be necessary to facilitate the disposition of all
              Registrable Securities covered by such Registration Statement
              during the applicable period in accordance with the intended
              method or methods of disposition thereof set forth in such
              Registration Statement or such prospectus or supplement thereto;

                        (ii) notify the Holders, promptly (A) when each
              Registration Statement, prospectus, or supplement thereto or
              further post-effective amendment has been filed, and, with
              respect to each Registration Statement or further post-effective
              amendment, when it has become effective, (B) of any request by
              the SEC for amendments or supplements to any Registration
              Statement or prospectus or for additional information, (C) of the
              issuance by the SEC of any comments with respect to any filing
              and of any stop order suspending the effectiveness of any
              Registration Statement or the initiation of any proceedings for
              that purpose, (D) of the receipt by the Company of any
              notification with respect to the suspension of the qualification
              of any Registrable Securities for sale in any jurisdiction or the
              initiation or threatening of any proceeding for such purpose, (E)
              of the happening of any event that makes any statement made in
              any Registration Statement, prospectus, or any other document
              incorporated therein by reference untrue or that requires the
              making of any changes in such Registration Statement, prospectus,
              or any document incorporated therein by reference in order that
              such documents not contain any untrue statement of a material
              fact or omit to state any material fact required to be stated
              therein or necessary to make the statements therein not
              misleading, and (F) of the Company's determination that a further
              post-effective amendment to such Registration Statement would be
              appropriate;

                       (iii) furnish to each Holder, without charge, as many
              conformed copies as may reasonably be requested by such Holder,
              of each Registration Statement and any further post-effective
              amendments thereto, including financial statements and schedules,
              all documents incorporated therein by reference and all exhibits
              (including those incorporated by reference);

                        (iv) deliver to each Holder, without charge, as many
              copies of the then effective prospectus covering such Registrable
              Securities and any amendments or supplements thereto as such
              Holder may reasonably request;

                         (v) register, qualify, obtain an exemption therefrom,
              or cooperate with the Holders and their counsel in connection
              with the registration or qualification or exemption therefrom of
              such Registrable Securities for offer and sale under the
              securities or blue sky laws of such jurisdictions as may be
              reasonably requested in writing by the Holders and do any and all
              other acts or things necessary or advisable to enable the
              disposition in such jurisdictions of the Registrable Securities
              covered by the then effective Registration Statements; provided,
              however, that the Company shall not be required to (A) qualify as
              a foreign corporation or generally to transact business in any
              jurisdiction where it is not then so qualified, (B) qualify as a
              dealer (or other similar entity) in securities, (C)


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              otherwise subject itself to taxation in connection with such
              activities, or (D) take any action which would subject it to
              general service of process in any jurisdiction where it is not
              then so subject;

                        (vi) upon the occurrence of any event contemplated by
              clauses (E) or (F) of paragraph (ii) above, promptly prepare and
              file, if necessary, a further post-effective amendment to each
              Registration Statement or a supplement to the related
              prospectuses or any document incorporated therein by reference or
              file any other required document so that each Registration
              Statement and the related prospectuses will not thereafter
              contain an untrue statement of a material fact or omit to state
              any material fact necessary to make the statements therein not
              misleading;

                       (vii) in no event later than five (5) Business Days
              before filing any Registration Statement, any further
              post-effective amendment thereto, any prospectus or any amendment
              or supplement thereto (other than any amendment or supplement
              made solely as a result of incorporation by reference of
              documents), furnish to the Holders copies of all such documents
              proposed to be filed;

                      (viii) not file any Registration Statement or amendment
              thereto or any prospectus or any supplement thereto (other than
              any amendment or supplement made solely as a result of
              incorporation by reference of documents) to which the Holders
              holding a majority of the Registrable Securities shall have
              reasonably objected in writing, within three (3) Business Days
              after receipt of such documents, to the effect that such
              Registration Statement or amendment thereto or prospectus or
              supplement thereto does not comply in all material respects with
              the requirements of the Securities Act (including, without
              limitation respect of any information describing the manner in
              which the Holders acquired such Registrable Securities and the
              intended method or methods of distribution of such Registrable
              Securities), (provided that the foregoing shall not limit the
              right of any Holder reasonably to object, within three (3)
              Business Days after receipt of such documents, to any particular
              information relating specifically to such Holder that is to be
              contained in any Registration Statement, prospectus or supplement
              including without limitation, any information describing the
              manner in which such Holder acquired such Registrable Securities
              and the intended method or methods of distribution of such
              Registrable Securities), and if the Company is unable to file any
              such document due to the objections of the Holders, the Company
              shall exert commercially reasonable efforts to cooperate with the
              Holders to prepare, as soon as practicable, a document that is
              responsive in all material respects to the reasonable objections
              of the Holders, provided however, that the Effectiveness Deadline
              (as defined in Section 2(b)) shall be extended by the period
              during which the Company is prevented from filing a Registration
              Statement or amendment thereto by reason of this paragraph
              3(a)(viii);

                        (ix) promptly after the filing of any document that is
              to be incorporated by reference into any Registration Statement
              or prospectus, provide copies of such document to the Holder;

                         (x) cause all Registrable Securities covered by each
              Registration Statement to be listed on the Nasdaq National
              Market, any national securities exchange, an over-the-



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              counter market, or, if the Common Stock securities of the Company
              are not listed thereon, on the primary exchanges, markets, or
              inter-dealer quotations systems (including NASDAQ) if any, on
              which similar securities issued by the Company are then listed,
              prior to the date on which such Registrable Securities were
              issued;

                        (xi) take all actions reasonably required to prevent
              the entry of any stop order by the Securities and Exchange
              Commission or by any state securities regulators or to remove any
              such order if entered; and

                       (xii) file post-effective amendments to any Registration
              Statement or supplement the related prospectus, as required, to
              permit sales of Registrable Securities covered thereby to be made
              by Permitted Transferees of the Holders.

              (b)     Holder Procedures.

                         (i) The Company may require each Holder to furnish to
              the Company such information regarding such Holder and the
              proposed distribution of such Registrable Securities as the
              Company may from time to time reasonably request in writing and
              which is necessary for compliance with applicable law.

                        (ii) Each Holder agrees to cooperate with the Company
              in all reasonable respects in connection with the preparation and
              filing of the Registration Statement, any Amendment, any
              prospectus, and any prospectus supplement.

              (c)     Additional Information Available. So long as any
Registration Statement is effective covering the resale of Registrable
Securities owned by a Holder, the Company will furnish to such Holder(s):

                         (i) as soon as practicable after it becomes available
              (but in the case of the Company's Annual Report to Stockholders,
              within 140 days after the end of each fiscal year of the
              Company), one copy of: (A) its Annual Report to Stockholders
              (which Annual Report shall contain financial statements audited
              in accordance with generally accepted accounting principles by a
              national firm of certified public accountants); (B) its Annual
              Report on Form 10-K; and (C) its Quarterly Reports on Form 10-Q;
              and

                        (ii) upon the reasonable request of a Holder, all
              exhibits to the Annual Report on Form 10-K; and the Company, upon
              the reasonable request of a Holder, will meet with such Holder or
              a representative thereof at the Company's headquarters to discuss
              all information relevant for disclosure in any Registration
              Statement and will otherwise cooperate with any Holder conducting
              an investigation for the purpose of reducing or eliminating such
              Holder's exposure to liability under the Securities Act including
              the reasonable production of information at the Company's
              headquarters.

4.            REGISTRATION EXPENSES.

              All expenses incident to the Company's performance of or
compliance with this Agreement, including without limitation all registration
and filing fees, fees, and expenses of


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compliance with state securities or blue sky laws (including fees and
disbursements of counsel in connection with blue sky qualifications or
registrations (or the obtaining of exemptions therefrom) of the Registrable
Securities), messenger and delivery expenses, internal expenses (including,
without limitation, all salaries and expenses of its officers and employees
performing legal or accounting duties), fees and disbursements of its counsel
and its independent certified public accountants, securities acts liability
insurance (if the Company elects to obtain such insurance), and reasonable fees
and expenses of any special experts retained by the Company in connection with
any registration hereunder (all of such expenses herein referred to as
"Registration Expenses"), shall be borne by the Company; provided, however, the
Registration Expenses shall not include any sales or underwriting discounts,
commissions, or fees attributable to the sale of the Registrable Securities or
the fees and expenses of counsel to the Holders.

5.            INDEMNIFICATION; CONTRIBUTION.

              (a)     Indemnification by the Company. The Company shall
indemnify and hold harmless, to the full extent permitted by law, each Holder,
and such Holder's respective officers, directors, employees, representatives,
agents, and controlling persons (within the meaning of the Securities Act),
against all losses, claims, damages, liabilities, and expenses resulting from
any untrue or alleged untrue statement of a material fact contained in any
Registration Statement, any prospectus, or any amendment or supplement thereto,
or any omission or alleged omission of a material fact required to be stated
therein or necessary to make the statements therein not misleading, except in
each case insofar as the same directly arises out of or is directly based upon
an untrue statement or alleged untrue statement of a material fact or an
omission or alleged omission to state a material fact in such Registration
Statement, prospectus, amendment, or supplement, as the case may be, made or
omitted, as the case may be, in reliance upon and in conformity with
information furnished to the Company in writing by such Holder expressly for
use therein.

              (b)     Indemnification by the Holders. Each Holder shall
indemnify and hold harmless, to the full extent permitted by law, the Company,
its officers, directors, employees, representatives, agents, and controlling
persons (within the meaning of the Securities Act), against all losses, claims,
damages, liabilities, and expenses (including, without limitation, reasonable
costs of investigation and legal expenses) resulting from any untrue or alleged
untrue statement of a material fact contained in any Registration Statement,
any prospectus, or any amendment or supplement thereto, and any omission or
alleged omission of a material fact required to be stated therein or necessary
to make the statements therein not misleading, to the extent the same directly
result from any untrue statement or alleged untrue statement of a material fact
or any omission or alleged omission to state a material fact in such
Registration Statement, prospectus, amendment, or supplement, as the case may
be, made or omitted, as the case may be, in reliance upon and in conformity
with information furnished to the Company in writing by such Holder expressly
for use therein. The liability of each Holder under the indemnity and
contributions provisions of this Section 5 shall be several and not joint and
shall be limited to an amount equal to the gross price of the Registrable
Securities sold by such Holder pursuant to the Registration Statement.

              (c)     Conduct of Indemnification Proceedings. Each party
entitled to indemnification under this Section 5 (the "Indemnified Party")
shall give written notice to the party required to provide indemnification (the
"Indemnifying Party") promptly after such Indemnified Party has actual
knowledge of any claim as to which indemnity may be sought, and shall permit
the Indemnifying Party



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<PAGE>   8





to assume the defense of any such claim or any litigation resulting therefrom;
provided, that counsel for the Indemnifying Party who will conduct the defense
of such claim or litigation, is approved by the Indemnified Party (whose
approval will not be unreasonably withheld or delayed); and provided, further,
that the failure of any Indemnified Party to give notice as provided herein
shall not relieve the Indemnifying Party of its obligations except to the
extent that its defense of the claim or litigation involved is prejudiced by
such failure. The Indemnified Party may participate in such defense at such
party's expense; provided, however, that the Indemnifying Party shall pay such
expense if representation of such Indemnified Party by the counsel retained by
the Indemnifying Party would be inappropriate due to actual or potential
conflicts of interest between the Indemnified Party and any other party
represented by such counsel in such proceeding. No Indemnifying Party, in the
defense of any such claim or litigation, except with the consent of each
Indemnified Party, shall consent to entry of any judgment or enter into any
settlement that does not include as an unconditional term thereof the giving by
the claimant or plaintiff to such Indemnified Party of a release from all
liability in respect of any claim or litigation, and no Indemnified Party will
consent to entry of any judgment or settle any claim or litigation without the
prior written consent of the Indemnifying Party (not to be unreasonably
withheld or delayed). Each Indemnified Party shall furnish such information
regarding himself, herself or itself and the claim in question as the
Indemnifying Party may reasonably request and as shall be reasonably required
in connection with the defense of such claim and litigation resulting
therefrom.

              (d)     Contribution.

                         (i) If for any reason the indemnification provided for
              in this Section 5 from an Indemnifying Party, although otherwise
              applicable by its terms, is unavailable to an Indemnified Party
              hereunder, then the Indemnifying Party, in lieu of indemnifying
              such Indemnified Party, shall contribute to the amount paid or
              payable by the Indemnified Parties as a result of such losses,
              claims, damages, liabilities, or expenses in such proportion as
              is appropriate to reflect the relative fault of such Indemnifying
              Party and the Indemnified Parties in connection with the actions
              that resulted in such losses, claims, damages, liabilities, or
              expenses, as well as any other relevant equitable considerations.
              The relative fault of such Indemnifying Party and the Indemnified
              Parties shall be determined by reference to, among other things,
              whether any action in question, including any untrue or alleged
              untrue statement of a material fact, has been made by, or relates
              to information supplied by, such Indemnifying Party or the
              Indemnified Parties, and the parties' relative intent, knowledge,
              access to information, and opportunity to correct or prevent such
              action. The amount paid or payable by a party as a result of the
              losses, claims, damages, liabilities, and expenses referred to
              above shall be deemed to include, subject to the limitations set
              forth in Section 5(c), any legal or other fees or expenses
              reasonably incurred by such party in connection with any
              investigation or proceeding.

                        (ii) The parties hereto agree that it would not be just
              and equitable if contribution pursuant to this Section 5(d) were
              determined by pro rata allocation or by any other method of
              allocation that does not take account of the equitable
              considerations referred to in the immediately preceding
              paragraph. No person guilty of fraudulent misrepresentation
              (within the meaning of Section 11(f) of the Securities Act) shall
              be entitled to contribution from any Person who was not guilty of
              such fraudulent misrepresentation.




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<PAGE>   9




6.            RULE 144 REQUIREMENTS.

              The Company agrees to:

              (a)     use its best efforts to make and keep public information
available, as those terms are understood and defined in Rule 144 under the
Securities Act;

              (b)     use its best efforts to file with the SEC in a timely
manner all reports and other documents required of the Company under the
Securities Act and the Securities Exchange Act; and

              (c)     furnish to each Holder upon request a written statement by
the Company as to its compliance with the reporting requirements of said Rule
144 and of the Securities Act and the Securities Exchange Act.

7.            INJUNCTIONS.

              Each of the parties hereto acknowledges and agrees that one or
more of the parties would be damaged irreparably in the event the provisions of
this Agreement are not performed in accordance with their specific terms or
otherwise are breached. Accordingly, each of the parties agrees that the other
party shall be entitled to an injunction or injunctions to prevent breaches of
the provisions hereof in any action instituted in any court of the United
States or any state thereof having jurisdiction over the parties in the matter,
in addition to any other remedy to which it may be entitled, at law or in
equity.

8.            TERMINATION.

              This Agreement shall terminate on the Registration Termination
Date; provided, however, that the provisions of Section 5, 6, and 7 shall
survive the termination of this Agreement.

9.            MISCELLANEOUS.

              (a)     Amendments and Waivers. Except as otherwise provided
herein, the provisions of this Agreement may not be amended, supplemented, or
otherwise modified, and waivers or consents to departures from the provisions
hereof may not be given, unless the Company shall have obtained the prior
written consent of the Holders holding more than 80% of the Registrable
Securities at the time of such amendment.

              (b)     Notices. Unless otherwise specified, all notices and other
communications provided for between the Company and the Purchaser in this
Agreement shall be in writing, including telecopy, and delivered or transmitted
to the addresses set forth below, or to such other address as shall be
designated by the Company or the Purchaser in written notice to the other
party. Notice sent by telecopy shall be deemed to be given and received when
receipt of such transmission is acknowledged, and delivered notice shall be
deemed to be given and received when receipted for by, or actually received by,
an authorized officer of the Company or the Purchaser, as the case may be.

                      (i) if to the Purchaser, to such address as is specified
              for the Purchaser under the terms of the Credit Agreement or if
              to another Holder, to the address indicated on the



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<PAGE>   10


              Company's register relating to the Warrants or Registrable
              Securities held by such Holder or at such other address as such
              Holder may have furnished to the Company in writing.

                      (ii)     if to the Company, at:

                                        Brigham Exploration Company
                                        6300 Bridge Point Parkway
                                        Building 2, Suite 500
                                        Austin, TX 78730
                                        Attn: Chief Financial Officer
                                        telephone:        512-427-3300
                                        telecopier:       512-427-3400

              (c)     Successors and Assigns. This Agreement shall be binding
upon and shall inure to the benefit of the Company and each Holder and their
respective successors and assigns. The rights provided by this Agreement shall
be transferred automatically from the Purchaser to any Person to whom any Note,
any Warrants, or any Registrable Securities are transferred, provided that (x)
the Company is given written notice of the transfer and the name, address,
telephone number, and facsimile number of the transferees and (y) the
subsequent transferee agrees in writing to be bound by all of the terms of this
Agreement (any transferee referred to in the foregoing clauses being referred
to herein as a "Permitted Transferee").

              (d)     Counterparts. This Agreement may be executed in any number
of counterparts and by the parties in separate counterparts, each of which when
so executed shall be deemed to be an original and all of which taken together
shall constitute one and the same agreement.

              (e)     Headings; Construction. The Section numbers and headings
in this Agreement are for convenience of reference only and shall not limit or
otherwise affect the meaning hereof. Unless the context otherwise requires, all
references to Sections are to sections of this Agreement, and words in the
singular include the plural and vice versa. In computing any period of time
specified in this Agreement or in any notices, the date of the act or event
from which such period of time is to be measured shall be included, any such
period shall expire at 5:00 p.m., Houston time, on the last day of such period,
and any such period denominated in months shall expire on the date in the last
month of such period that has the same numerical designation as the date of the
act or event from which such period is to be measured; provided, however, that
if there is no date in the last month of such period that has the same
numerical designation as of the date of such act or event, such period shall
expire on the last day of the last month of such period.

              (f)     Governing Law. This Agreement shall be governed by and
construed in accordance with the internal laws of Texas, without regard to the
principles of conflicts of laws thereof which would select any other law.

              (g)     Severability. If one or more of the provisions hereof, or
the application thereof in any circumstance, is held invalid, illegal, or
unenforceable in any respect, for any reason, the validity, legality, and
enforceability of the remaining provisions hereof shall not be in any way
affected or impaired thereby, and the provisions held to be invalid, illegal,
or unenforceable shall be reformed


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<PAGE>   11



to the minimum extent necessary, and in a manner as consistent with the
purposes thereof as is practicable, so as to render it valid, legal, and
enforceable.

              (h)     Entire Agreement. This Agreement, the Credit Agreement,
the Loan Documents, the Equity Conversion Agreement and the Warrant Agreement
are intended by the parties hereto to be a final expression thereof and is
intended to be a complete and exclusive statement of the agreement and
understanding of such parties in respect of the subject matter contained herein
related to the Registrable Securities. These agreements supersede all prior
agreements and understandings among the Company and any of the Holders with
respect to such subject matter.



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<PAGE>   12





              IN WITNESS WHEREOF, the parties hereto have caused this Agreement
to be duly executed as of the day and year first above written.


                            BRIGHAM EXPLORATION COMPANY


                            By:
                                 ----------------------------------------------
                            Name:  Curtis F. Harrell
                            Title: Chief Financial Officer


                            SHELL CAPITAL INC.


                            By:
                                 ----------------------------------------------
                            Name:  Robert L. Roberts
                            Title: Vice-President



[Registration Rights Agreement]