**Exhibit 4.1****THE HOUSTON EXPLORATION COMPANY**AND**THE BANK OF NEW YORK,**as Trustee**FIRST SUPPLEMENTAL INDENTURE**Dated as of May 21, 2007toIndentureDated as of June 10, 20037% Senior Subordinated Notes due 2013
 

 
     **THIS FIRST SUPPLEMENTAL INDENTURE**(this “***Supplemental Indenture***”), dated as of May 21, 2007,is by and between The Houston Exploration Company, a Delaware corporation (the “***Company***”), and TheBank of New York, a New York banking corporation, as trustee (the “***Trustee***”).     WHEREAS, the Trustee and the Company have heretofore executed and delivered that certainIndenture dated as of June 10, 2003 (as amended, supplemented or otherwise modified from time totime, the “***Indenture***”), providing for the issuance of the Company’s 7% Senior Subordinated Notesdue 2013 (the “***Notes***”);     WHEREAS, on June 10, 2003, the Company issued $175,000,000 aggregate principal amount ofRestricted Notes (as defined in the Indenture), and it subsequently exchanged all of such Notes forUnrestricted Notes pursuant to the Exchange Offer (as such terms are defined in the Indenture), allof which Unrestricted Notes are currently outstanding;     WHEREAS, Section 9.2 of the Indenture provides that, with the consent of Holders representinga majority in principal amount of the Notes then outstanding, the Company, and the Trustee mayenter into an indenture supplemental to the Indenture for the purpose of amending or supplementingthe Indenture or the Notes (subject to certain exceptions);     WHEREAS, the Company desires and has requested the Trustee to join with it in entering intothis Supplemental Indenture for the purpose of amending the Indenture and the Notes in certainrespects as permitted by Section 9.2 of the Indenture;     WHEREAS, the Company has been soliciting consents to this Supplemental Indenture upon theterms and subject to the conditions set forth in its Offer to Purchase and Consent SolicitationStatement dated May 2, 2007 (which statement, including any amendments, modifications orsupplements thereto, constitutes the “***Tender Offer***”);     WHEREAS, there are no Subsidiary Guarantors;     WHEREAS, pursuant to Section 9.2 of the Indenture, the Company has delivered a request to theTrustee, accompanied by a resolution of its Board of Directors authorizing the execution of thisSupplemental Indenture, requesting the Trustee to join with the Company in the execution of thisSupplemental Indenture;     WHEREAS, pursuant to Sections 9.2 and 9.6 of the Indenture, the Board of Directors hasauthorized and approved this Supplemental Indenture; and     WHEREAS, (1) the Company has received the consent of the Holders of a majority in principalamount of the outstanding Notes to the substance of this Supplemental Indenture, evidence of whichthat is satisfactory to the Trustee having been filed therewith, all as certified by an Officers’Certificate delivered to the Trustee simultaneously with the execution and delivery of thisSupplemental Indenture, (2) the Company has delivered to the Trustee simultaneously with theexecution and delivery of this Supplemental Indenture an Officers’ Certificate and an Opinion ofCounsel relating to this Supplemental Indenture as contemplated by Sections 9.6 and 12.4 of theIndenture and (3) the Company has satisfied all other conditionsExhibit A
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required under the Indenture to enable the Company and the Trustee to enter into thisSupplemental Indenture.     NOW, THEREFORE, in consideration of the above premises, each party hereby agrees, for thebenefit of the others and for the equal and ratable benefit of the Holders of the Notes, asfollows:**ARTICLE I**