**Exhibit 4.1**
**FIRST AMENDMENT TO****RIGHTS AGREEMENT**
     This First Amendment to the Rights Agreement (the “*Amendment*”), is made and entered into as ofMay 2, 2005, between The Houston Exploration Company, a Delaware corporation (the “*Company*”), andThe Bank of New York, as Rights Agent (the “*Rights Agent*”).
W I T N E S S E T H:
     WHEREAS, the Company and the Rights Agent have determined that it would be in the bestinterests of the Company and the Rights Agent to amend certain provisions of the Rights Agreement,dated as of August 12, 2004, between the Company and the Rights Agent (the “*Agreement*”); and
     NOW, THEREFORE, in consideration of the foregoing, the terms and conditions contained herein,and other good and valuable consideration, the receipt and sufficiency of which are herebyacknowledged, the parties hereto agree as follows:
     1. Defined Terms. Capitalized terms used herein but not defined herein shall have therespective meanings set forth in the Agreement.
     2. Amendments to Section 1. The definitions in Section 1 (“Certain Definitions”) for“Acquiring Person” and “Exempt Person” are hereby amended and restated in their entirety asfollows:
     (a) “(a) “Acquiring Person” shall mean any Person (as such term is hereinafter defined) who orwhich shall be the Beneficial Owner (as such term is hereinafter defined) of 15% or more of theshares of Common Stock then outstanding, but shall not include an Exempt Person (as such term ishereinafter defined); provided, however, that (i) if the Board of Directors of the Companydetermines in good faith that a Person who would otherwise be an “Acquiring Person” became theBeneficial Owner of a number of shares of Common Stock such that the Person would otherwise qualifyas an “Acquiring Person” inadvertently (including, without limitation, because (A) such Person wasunaware that it beneficially owned a percentage of Common Stock that would otherwise cause suchPerson to be an “Acquiring Person” or (B) such Person was aware of the extent of its BeneficialOwnership of Common Stock but had no actual knowledge of the consequences of such BeneficialOwnership under this Agreement) and without any intention of changing or influencing control of theCompany, then such Person shall not be deemed to be or to have become an “Acquiring Person” for anypurposes of this Agreement unless and until such Person shall have failed to divest itself, as soonas practicable (as determined, in good faith, by the Board of Directors of the Company), ofBeneficial Ownership of a sufficient number of shares of Common Stock so that such Person would nolonger otherwise qualify as an “Acquiring Person”; (ii) if, as of the date hereof or prior to thefirst public announcement of the adoption of this Agreement, any Person is or becomes theBeneficial Owner of 15% or more of the shares of Common Stock outstanding, such Person shall not bedeemed to be or to become an “Acquiring Person” unless and until such time as such Person shall,after the first public announcement of the adoption of this Agreement, become the Beneficial Ownerof additional shares of Common Stock (other than pursuant to a dividend or distribution paid ormade by the Company on the outstanding Common Stock or pursuant to a split or subdivision of theoutstanding Common Stock), unless, upon becoming the Beneficial Owner of such additional shares ofCommon Stock, such Person is not then the Beneficial Owner of 15% or more of the shares of CommonStock then outstanding; and (iii) no Person shall become an “Acquiring Person” as the result of anacquisition of shares of Common Stock by the Company which, by reducing the number of sharesoutstanding, increases the proportionate number of shares of Common Stock beneficially owned bysuch Person to 15% or more of the shares of Common Stock then outstanding, provided, however, thatif a Person shall become the Beneficial Owner of 15% or more of the shares of Common Stock thenoutstanding by reason of such share acquisitions by the Company and shall thereafter become theBeneficial Owner of any additional shares of Common Stock (other than pursuant to a dividend ordistribution paid or made by the Company on the outstanding Common Stock or pursuant to a split orsubdivision of the outstanding Common Stock), then such Person shall be deemed to be an “Acquiring
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Person” unless upon becoming the Beneficial Owner of such additional shares of Common Stock suchPerson does not beneficially own 15% or more of the shares of Common Stock then outstanding. Forall purposes of this Agreement, any calculation of the number of shares of Common Stock outstandingat any particular time, including for purposes of determining the particular percentage of suchoutstanding shares of Common Stock of which any Person is the Beneficial Owner, shall be made inaccordance with the last sentence of Rule 13d-3(d)(1)(i) of the General Rules and Regulations underthe Securities Exchange Act of 1934, as amended (the “Exchange Act”), as in effect on the datehereof.”
     (b) “(k) “Exempt Person” shall mean the Company or any Subsidiary (as such term is hereinafterdefined) of the Company, in each case including, without limitation, in its fiduciary capacity, orany employee benefit plan of the Company or of any Subsidiary of the Company, or any entity ortrustee holding Common Stock for or pursuant to the terms of any such plan or for the purpose offunding any such plan or funding other employee benefits for employees of the Company or of anySubsidiary of the Company.”
     3. Amendment to Exhibit C. The second paragraph of Exhibit C (“Summary of Rights toPurchase Shares of Preferred Stock of The Houston Exploration Company”) of the Agreement is herebyamended and restated in its entirety as follows:
     “Until the earlier to occur of (i) 10 days following a public announcement that a person orgroup of affiliated or associated persons has become an Acquiring Person (as defined below) or (ii)10 business days (or such later date as may be determined by action of the Board of Directors ofthe Company prior to such time as any person or group of affiliated persons becomes an AcquiringPerson) following the commencement of, or announcement of an intention to make, a tender offer orexchange offer the consummation of which would result in any person or group of affiliated personbecoming an Acquiring Person (the earlier of such dates being called the “Distribution Date”), theRights will be evidenced, with respect to any of the Common Stock certificates outstanding as ofthe Record Date, by such Common Stock certificate together with this Summary of Rights. Thedetermination of whether and when a person or group of affiliated or associated persons becomes an“Acquiring Person” shall be made in accordance with the terms and provisions of the RightsAgreement.”
     4. Effect of Amendment. Except as expressly set forth herein, the terms andprovisions of the Agreement remain in full force and effect and are hereby ratified and confirmed.References in any agreement, instrument or other document to the Agreement shall be deemed to be areference to the Agreement as amended hereby.
     5. Effectiveness Date. This Amendment shall be effective as of May 2, 2005.
     6. Governing Law. This Amendment shall be governed by the laws of the State ofDelaware.
     7. Counterparts. This Amendment may be executed in one or more counterparts, each ofwhich shall be deemed an original for all purposes and all of which together shall constitute oneand the same agreement.
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     IN WITNESS WHEREOF, each of the parties hereto has caused this Amendment to be executed anddelivered by its duly authorized officer as of the date first above written.         THE HOUSTON EXPLORATION COMPANY,