Exhibit 10.4
AMENDMENT
TO THE
HOUSTON EXPLORATION COMPANY
DEFERRED COMPENSATION PLAN
FOR
NON-EMPLOYEE DIRECTORS
     The Houston Exploration Company (the “Company”) wishes to amend the Houston
Exploration Company Deferred Compensation Plan for Non-Employee Directors (the
“Plan”) to freeze the Plan as of December 31, 2004 with respect to the value of
participants’ Plan accounts as of December 31, 2004 and any deemed earnings or
losses thereon accruing after December 31, 2004. The purpose of this Amendment
is to permit Plan participants to enjoy a grandfathering exemption from section
409A of the Internal Revenue Code of 1986, as amended, as provided under the
American Jobs Creation Act of 2004 and the authoritative guidance issued
thereunder (“AJCA”). No modification, whether in the form of a Plan amendment,
the exercise of any discretionary authority under the Plan, or otherwise, may be
made to the Plan which constitutes a “material modification” under AJCA, and any
attempt to do so will be void and without effect.
     Accordingly, effective December 31, 2004, the Plan is amended as follows:
     1.     A new Section 3.2 is added to read as follows:
               3.2     FREEZE OF PARTICIPATION. Notwithstanding any other
provision of this Plan, no individual shall become a Participant in this Plan
after December 31, 2004.
     2.     The following sentence is added to the end of Section 4.1:
               “Notwithstanding any other provision of this Plan, no
Compensation deferrals shall be deferred under this Plan after December 31,
2004.”
     3.     The following sentence is added to the end of Section 5.4:
               “Any such amendment and/or any other election taken with respect
to the Plan that constitutes a “material modification” of the Plan as defined in
the American Jobs Creation Act of 2004, and the authoritative guidance issued
thereunder, shall be void and without effect.”
     IN WITNESS WHEREOF, the Company has caused this Amendment to be executed
effective as of December 31, 2004.

                         
 
                        ATTEST/WITNESS:       THE HOUSTON EXPLORATION COMPANY
 
                        By:   /s/ Liz Melton       By:  /s/ Roger B. Rice      
        Print Name:  Liz Melton       Print Name:  Roger B. Rice                
Title:     Vice President                 Date:     4/26/05