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                                                                   Exhibit 4.2.1

                              [filed March 2, 2001]

                            CERTIFICATE OF AMENDMENT
                                       of
                           CERTIFICATE OF DESIGNATIONS

                                       of

                            SERIES A PREFERRED STOCK
                           (Par Value $.01 Per Share)
                                       of
                           BRIGHAM EXPLORATION COMPANY

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

                             Pursuant to Section 242
             of the General Corporation Law of the State of Delaware

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

      Brigham Exploration Company, a corporation organized and existing under
the laws of the State of Delaware (the "Corporation"), DOES HEREBY CERTIFY as
follows:

      FIRST: That the Board of Directors of the Corporation, by a unanimous
written consent in lieu of special meeting dated March 1, 2001, duly adopted
resolutions approving and adopting the amendments set forth below to the
Certificate of Designations of Series A Preferred Stock of the Corporation (the
"Certificate of Designations"), and that such amendments have been approved and
adopted by all existing holders of the Series A Preferred Stock of the
Corporation.

      SECOND: That the amendments set forth below have been duly adopted in
accordance with the applicable provisions of Section 242 of the General
Corporation Law of the State of Delaware.

      THIRD: That the Certificate of Designations is hereby amended as follows:

      A. Section 1 of the Certificate of Designations is hereby replaced in its
entirety with the following:

            "Section 1. Designation and Number of Shares of Series A Preferred
      Stock. There is hereby authorized and established a series of Preferred
      Stock that shall be designated as "Series A Preferred Stock" (hereinafter
      referred to as "Series A Preferred"), and the number of shares
      constituting such series shall be 2,250,000. Such number of shares may be
      increased or decreased, but not to a number less than the number of shares
      of Series A Preferred then issued and outstanding, by resolution adopted
      by the full Board of Directors. The "Stated Value" per share of the Series
      A Preferred shall be equal to Twenty Dollars ($20.00)."
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      B. The definition of "Warrants" contained in Section 2 of the Certificate
of Designations is hereby replaced in its entirety with the following:

            " "Warrants" means (i) the warrants to purchase Common Stock
      originally issued to DLJ MB Funding III, Inc. and DLJ ESC II, LP on
      November 1, 2000, pursuant to warrant certificates in the form attached
      hereto as Exhibit A and/or (ii) the warrants to purchase Common Stock
      originally issued to DLJ Merchant Banking Partners III, L.P., DLJ Offshore
      Partners III, C.V., DLJ MB Funding III, Inc. and DLJ ESC II, LP on or
      about March 5, 2001, pursuant to warrant certificates in the form attached
      hereto as Exhibit B."

      C. An Exhibit B to the Certificate of Designations is hereby added
thereto, in the form attached hereto as Exhibit B.

      IN WITNESS WHEREOF, this Certificate of Amendment has been duly executed
on behalf of the Corporation by its authorized officer, this 2nd day of March,
2001.

                                           BRIGHAM EXPLORATION COMPANY


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