SUPPLEMENTAL INDENTURE NUMBER 2

Exhibit 4.15

SUPPLEMENTAL INDENTURE NUMBER 2

Dated as of June 1, 2003

to

INDENTURE

Dated as of October 27, 1998

between

ACE US HOLDINGS, INC.,

as Issuer

and

THE BANK OF NEW YORK,

as Successor Trustee

SUPPLEMENTAL INDENTURE NUMBER 2

This SUPPLEMENTAL INDENTURE NUMBER 2, dated as of June 1, 2003, is made among ACE US Holdings, Inc., a Delaware corporation (the “Company”) and THE BANK OF NEW YORK, as successor trustee under the indenture referred to below (the “Trustee”).

WITNESSETH:

WHEREAS, the Company has heretofore executed and delivered to the United States Trust Company of New York, as trustee, an Indenture, dated as of October 27, 1998, as amended by a Supplemental Indenture No. 1 and Waiver dated February 16, 2000 (as so amended, the “Indenture”) providing for the issuance of an aggregate principal amount of up to $250,000,000 of Credit Sensitive Senior Notes due 2008 (the “Securities”);

WHEREAS, United States Trust Company of New York has transferred all or substantially all of its corporate trust business or assets to The Bank of New York and, pursuant to Section 7.09 of the Indenture, The Bank of New York has become the successor trustee;

WHEREAS, Section 4.02(d) of the Indenture provides that within 45 days of the end of each fiscal quarter, the Company will provide to the Trustee, the Calculation Agent and the Securityholders an Officers’ Certificate (the “Quarterly Certificate”) containing certain financial calculations;

WHEREAS, a defect is contained in said Section 4.02(d) in that the Quarterly Certificate with respect to the last fiscal quarter of each year should be provided within 120 days of the end of each such fiscal quarter and not 45 days as contained in the Indenture;

WHEREAS, the Issuer has requested, and the Trustee is prepared to agree to, amend Section 4.02(d) of the Indenture to cure such defect (the “Amendment”);

WHEREAS, pursuant to Section 9.01 of the Indenture, the Trustee and the Company are authorized to execute and deliver this Supplemental Indenture Number 2;

NOW THEREFORE, in consideration of the foregoing and for other good and valuable consideration, the receipt of which is hereby acknowledged, the Company and the Trustee mutually covenant and agree for the equal and ratable benefit of the Securityholders as follows:

1. Amendment of Section 4.02(d) of the Indenture. Section 4.02(d) of the Indenture is hereby replaced in its entirety with the following:

(d) The Company shall deliver to the Trustee, the Calculation Agent and the Securityholders, no later than (i) in the case of the first three fiscal quarters of each year, the 45th day following the last day of such fiscal quarters and (ii) in the case of the last fiscal quarter of each year, the 120^th day following

the last day of such fiscal quarter, a certificate signed by two Officers of the Company (the “Quarterly Certificate”) in the form attached hereto as Exhibit D setting forth the following information: (i) the calculation of the Company’s Indebtedness/Total Capital Ratio as of the last day of such preceding fiscal quarter, measured in accordance with GAAP on a Consolidated basis, (ii) the current S&P Rating or Moody’s Rating for each ACE USA Insurance Group as of the date of such Quarterly Certificate, and (iii) based upon the calculation and rating set forth respectively in (i) and (ii) above, the applicable interest rate in effect for the next Pricing Period; and

2. Ratification of Indenture; Supplemental Indentures Part of Indenture. Except as expressly amended hereby, the Indenture is in all respects ratified and confirmed and all the terms, conditions and provisions thereof shall remain in full force and effect. This Supplemental Indenture Number 2 shall form a part of the Indenture for all purposes, and every Securityholder heretofore or hereafter authenticated and delivered shall be bound hereby.

3. Defined Terms. Capitalized terms used herein without definition shall have the respective terms assigned such terms in the Indenture.

4. Governing Law. THIS SUPPLEMENTAL INDENTURE NUMBER 2 SHALL BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE STATE OF NEW YORK (WITHOUT REGARD TO THE CHOICE OF LAW PROVISIONS THEREOF THAT WOULD REQUIRE THE APPLICATION OF ANY OTHER LAWS).

5. Trustee Makes No Representation. The Trustee makes no representation as to the validity or sufficiency of this Supplemental Indenture Number 2.

6. Counterparts. The parties may sign any number of copies of this Supplemental Indenture Number 2. Each signed copy shall be an original, but all of them together represent the same agreement.

7. Effect of Headings. The Section headings herein are for convenience only and shall not effect the construction thereof.

2

IN WITNESS WHEREOF, the parties hereto have caused this Supplemental Indenture Number 1 to be duly executed as of the date first above written.

3