Exhibit 31.1

                    CERTIFICATION OF CHIEF EXECUTIVE OFFICER
                        PURSUANT TO RULE 13A-14(a) OF THE
                         SECURITIES EXCHANGE ACT OF 1934

I, Ben M. Brigham, certify that:

1.   I have reviewed this quarterly report on Form 10-Q of Brigham
     Exploration Company;

2.   Based on my knowledge, this report does not contain any untrue
     statement of a material fact or omit to state a material fact necessary to
     make the statements made, in light of the circumstances under which such
     statements were made, not misleading with respect to the period covered by
     this report;

3.   Based on my knowledge, the financial statements, and other financial
     information included in this report, fairly present in all material
     respects the financial condition, results of operations and cash flows of
     the registrant as of, and for, the periods presented in this report;

4.   The registrant's other certifying officer(s) and I are responsible for
     establishing and maintaining disclosure controls and procedures (as defined
     in Exchange Act Rules 13a-15e and 15d-15(e)) for the registrant and have:

     a)   Designed such disclosure controls and procedures or caused such
          disclosure controls and procedures to be designed under our
          supervision, to ensure that material information relating to the
          registrant, including its consolidated subsidiaries, is made known to
          us by others within those entities, particularly during the period in
          which this report is being prepared;

     b)   Evaluated the effectiveness of the registrant's disclosure
          controls and procedures and presented in this report our conclusions
          about the effectiveness of the disclosure controls and procedures, as
          of the end of the period covered by this report based on such
          evaluation; and

     c)   Disclosed in this report any change in the registrant's internal
          control over financial reporting that occurred during the registrant's
          most recent fiscal quarter (the registrant's fourth fiscal quarter in
          the case of an annual report) that has materially affected, or is
          reasonably likely to materially affect, the registrant's internal
          control over financial reporting; and

5.   The registrant's other certifying officer(s) and I have disclosed,
     based on our most recent evaluation of internal control over financial
     reporting, to the registrant's auditors and the audit committee of the
     registrant's board of directors (or persons performing the equivalent
     functions):

     a)   All significant deficiencies and material weaknesses in the
          design or operation of internal control over financial reporting,
          which are reasonably likely to adversely affect the registrant's
          ability to record, process, summarize and report financial
          information; and

     b)   Any fraud, whether or not material, that involves management or
          other employees who have a significant role in the registrant's
          internal control over financial reporting.


Date:  May 6, 2005

           /s/ BEN M. Brigham
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           Ben M. Brigham
           Chief Executive Officer, President and
           Chairman of the Board