<PAGE>

                                                                   Exhibit 10.71

                               First Amendment to
                          REGISTRATION RIGHTS AGREEMENT

      THIS FIRST AMENDMENT TO REGISTRATION RIGHTS AGREEMENT (this "Agreement")
is entered into as of March 5, 2001, by and among Brigham Exploration Company, a
Delaware corporation (the "Company"), and the parties listed on Schedule A
hereto.

                                    RECITALS

      WHEREAS, pursuant to that certain Registration Rights Agreement dated as
of November 1, 2000 (the "Original Agreement"), by and among the Company and
DLJMB Funding III, Inc., a Delaware corporation and DLJ ESC II, L.P., a Delaware
Limited Partnership (the "Initial Investors"), the Company provided certain
registration rights to the Initial Investors regarding shares of the Company's
common stock issuable upon conversion of warrants issued by the Company to the
Initial Investors;

      WHEREAS, each of the Initial Investors and DLJ Merchant Banking Partners
III, L.P., a Delaware Limited Partnership, and DLJ Offshore Partners III, C.V.,
a Netherlands Antilles Limited Partnership (collectively hereinafter referred to
as the "Investors"), and the Company are parties to that certain Securities
Purchase Agreement dated as of March 5, 2001 (the "New Securities Purchase
Agreement");

      WHEREAS, each of the Investors is a party to a certain Warrant Certificate
(the "New Warrant Certificate") dated as of March 5, 2001, by and between the
Company and each of such Investors;

      WHEREAS, the New Warrant Certificates were executed and delivered in
connection with the consummation of transactions contemplated by the new
Securities Purchase Agreement;

      WHEREAS, pursuant to the New Warrant Certificates, each of the Investors
has been issued a warrant (the "New Warrant") to purchase shares of the
Company's common stock, par value $.01 per share; and

      WHEREAS, to induce the Investors to enter into the New Warrant
Certificates and the new Securities Purchase Agreement, the Company has agreed
to provide registration rights under the Original Agreement with respect to the
shares issuable upon exercise of the New Warrants.

      NOW, THEREFORE, for and in consideration of the foregoing, and other good
and valuable consideration, the sufficiency of which are hereby acknowledged,
the parties hereto, intending to be legally bound, do hereby agree as follows:

Section 1. Amendments to the Original Agreement.

(a) The "RECITALS" section of the Original Agreement is amended by deleting such
section entirely and replacing it with the following:

            " WHEREAS, each of DLJMB Funding III, Inc. ("MB") and DLJ ESC II,
      L.P. ("ESC") (MB and ESC are sometimes hereinafter referred to
      collectively as the "Initial Investors") is a party to a certain Warrant
      Certificate (the "Warrant Certificate") dated as of November 1, 2000, by
      and between the Company and such Investor, and each of the Initial
<PAGE>

      Investors and DLJ Merchant Banking Partners III, L.P. ("MBP") and DLJ
      Offshore Partners III, C.V. ("Offshore") (the Initial Investors, MBP and
      Offshore are sometimes hereinafter referred to collectively as the
      "Investors") is a party to a certain Warrant Certificate (the "New Warrant
      Certificate") dated as of March 5, 2001, by and between the Company and
      such Investor;

            WHEREAS, the Warrant Certificates were executed and delivered to the
      Initial Investors in connection with the consummation of transactions
      contemplated by that certain Securities Purchase Agreement dated as of
      November 1, 2000, by and among the Company and the Initial Investors, and
      the New Warrant Certificates were executed and delivered in connection
      with the consummation of transactions contemplated by that certain
      Securities Purchase Agreement dated as of March 5, 2001 by and among the
      Company and the Investors;

            WHEREAS, pursuant to the Warrant Certificates, each Holder (as
      defined below) has been issued a warrant (the "Warrant") to purchase
      shares of the Company's common stock, par value $.01 per share (the
      "Company's Common Stock"), and pursuant to the New Warrant Certificates,
      each Holder (as defined below) has been issued a warrant (the "New
      Warrant") to purchase shares of the Company's Common Stock;

            WHEREAS, to induce the Initial Investors to enter into the Warrant
      Certificates, and the Investors to enter into the New Warrant Certificates
      and the Securities Purchase Agreements described above, the Company has
      agreed to grant the registration and other rights contained in this
      Agreement;"

(b) The definition of Registrable Securities in Section 1.(d) of the Original
Agreement is amended by deleting such Section 1.(d) entirely and replacing it
with the following:

            " (d) "Registrable Securities" means (i) the Common Stock of the
      Company issuable or issued upon exercise of the Warrants or the New
      Warrants and (ii) any Common Stock of the Company issued as (or issuable
      upon the conversion or exercise of any warrant, right or other security
      which is issued as) a dividend or other distribution with respect to, or
      in exchange for or in replacement of, such Warrants, New Warrants or
      Common Stock, excluding in all cases, however, any Registrable Securities
      sold by a person in a transaction in which its rights under this Agreement
      are not assigned; and"

(c) Schedule A to the Original Agreement is amended by deleting such Schedule A
entirely and replacing it with Schedule A to this Agreement.

      Section 2. No Other Changes. Except as explicitly amended by this
Amendment, the terms, conditions, rights and obligations under the Original
Agreement shall remain in full force and effect.

      Section 3. Counterparts. This Amendment may be executed by the parties
hereto in any number of counterparts, each of which shall be deemed an original,
but all of which shall constitute one and the same agreement.


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<PAGE>

      IN WITNESS WHEREOF, the parties hereto have executed this Amendment as of
the date set forth above.

                                        BRIGHAM EXPLORATION COMPANY

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

                                        Address:
                                        Brigham Exploration Company
                                        6300 Bridge Point Parkway
                                        Building 2, Suite 500
                                        Austin, TX  78730
                                        Attention: Chief Financial Officer
                                        Fax:  512-427-3400


                                        DLJMB FUNDING III, INC.

                                        /s/ Steven A. Webster
                                        ----------------------------------------
                                        Name:  Steven A. Webster
                                        Title:    Attorney-in-Fact

                                        Address:
                                        277 Park Avenue
                                        New York, New York 10172
                                        Attention: Nicole Arnaboldi, Ivy Dodes
                                        Telefax: 212-892-7272


                                        DLJ ESC II, L.P.

                                        By: DLJ LBO Plans Management Corporation
                                        ----------------------------------------
                                        General Partner

                                        /s/ Steven A. Webster
                                        ----------------------------------------
                                        Name:  Steven A. Webster
                                        Title:    Attorney-in-Fact

                                        Address:
                                        c/o DLJ LBO Plans Management Corporation
                                        277 Park Avenue
                                        New York, New York 10172
                                        Attention: Ivy Dodes/Nicole Arnaboldi
                                        Telefax: 212-892-7272


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<PAGE>

                                        DLJ Merchant Banking Partners III, L.P.

                                        By: DLJ Merchant Banking III, Inc.
                                        Managing General Partner

                                        /s/ Steven A. Webster
                                        ----------------------------------------
                                        Name:  Steven A. Webster
                                        Title: Attorney-in-Fact

                                        Address:
                                        DLJ Merchant Banking Partners III, L.P.
                                        277 Park Avenue
                                        New York, New York 10072
                                        Attention: Nicole Arnaboldi, Ivy Dodes
                                        Telefax: 212-892-7272

                                        DLJ Offshore Partners III, C.V.

                                        By: DLJ Merchant Banking III, Inc.
                                        Managing General Partner

                                        /s/ Steven A. Webster
                                        ----------------------------------------
                                        Name:  Steven A. Webster
                                        Title: Attorney-in-Fact

                                        Address:
                                        c/o DLJ Offshore Management N.V.
                                        John B. Gorsiraweg 14
                                        Willemstad, Curacao
                                        Netherlands, Antilles
                                        Fax:  011-599-961-4129


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                                   Schedule A
                        to Registration Rights Agreement

                                                      Number of Shares
Holder Name                                    November 1, 2000  March 5, 2001
-----------                                    ----------------  -------------

DLJ MB Funding III, Inc.,                         6,036,667         141,869
a Delaware corporation

DLJ ESC II, LP,                                     630,000         371,789
a Delaware limited partnership

DLJ Merchant Banking Partners III, L.P.                 -0-       1,527,154

DLJ Offshore Partners III, C.V                          -0-          64,451
                                                  ---------       ---------

      TOTAL                                       6,666,667       2,105,263


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