<PAGE>   1
                                                                 EXHIBIT 10.39.1


                   AMENDMENT TO REGISTRATION RIGHTS AGREEMENT


         This AMENDMENT TO REGISTRATION RIGHTS AGREEMENT (the "Amendment")
dated as of March 26, 1999, among Brigham Exploration Company, a Delaware
corporation (the "Company"), Joint Energy Development Investments II Limited
Partnership, a Delaware limited partnership ("JEDI-II"), and ECT Merchant
Investments Corp. ("ECT Merchant"), amends the Registration Rights Agreement
dated as of August 20, 1998 among the Company, JEDI-II and Enron Capital &
Trade Resources Corp., a Delaware corporation ("ECT").

         NOW, THEREFORE, in consideration of the mutual covenants and
agreements set forth herein and for good and valuable consideration, the
receipt of which is hereby acknowledged, the parties agree as follows:

                                    ARTICLE I
            MODIFICATION OF DEFINITION OF WARRANTS AND WARRANT SHARES

         Section 1.01. Definition of Warrant. "Warrants" means the Warrants
issued by the Company to JEDI-II and ECT Merchant as of the date hereof for the
purchase of an aggregate of 1,000,000 shares of Common Stock, evidenced as of
the date hereof by Warrant Certificates Nos. A-3 and A-4, and any Warrants
issued upon the transfer thereof or in substitution therefor, as the same may
be amended from time to time.

         Section 1.02. Definition of Warrant Shares. "Warrant Shares" shall mean
the shares of Common Stock and other securities receivable upon exercise of the
Warrants.

                                   ARTICLE II
                                  MISCELLANEOUS

         Section 2.01 Execution in Counterparts. This Amendment may be executed
in any number of counterparts and by different parties on separate counterparts
and all of such counterparts shall together constitute one and the same
instrument.

         Section 2.02 Governing Law. This Amendment and the rights and
obligations of the parties hereunder shall be construed in accordance with and
be governed by the laws of the State of Texas and the United States of America.

         Section 2.01 Descriptive Headings, etc. The descriptive headings of
the sections of this Amendment are inserted for convenience only and shall not
be deemed to affect the meaning or construction of any of the provisions
hereof.



<PAGE>   2



         IN WITNESS WHEREOF, the parties have executed this Amendment to
Registration Rights Agreement as of the date first written above.

                                 BRIGHAM EXPLORATION COMPANY



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

                                 JOINT ENERGY DEVELOPMENT
                                 INVESTMENTS II LIMITED PARTNERSHIP

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

                                     By:  Enron Capital II Corp., its General 
                                          Partner


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

                                 ECT MERCHANT INVESTMENTS CORP.


                                 By:      /s/ Mark Warner
                                          --------------------------------------
                                 Name:    Mark Warner 
                                          --------------------------------------
                                 Title:   Agent and Attorney in Fact
                                          --------------------------------------



                                      -2-