UNITED STATES

SECURITIES AND EXCHANGE COMMISSION

Washington, D.C. 20549

SCHEDULE 13G

o

Rule 13d-1(b)

ý

Rule 13d-1(c)

o

Rule 13d-1(d)

CUSIP No. G0450A105

1.

Names of Reporting Persons. I.R.S. Identification Nos. of above persons (entities only)
Cascade Investment, L.L.C.

2.

Check the Appropriate Box if a Member of a Group (See Instructions)

(a)

o

(b)

o

3.

SEC Use Only

4.

Citizenship or Place of Organization
State of Washington

Number of
Shares
Beneficially
Owned by
Each
Reporting
Person With

5.

Sole Voting Power
2,368,653*

6.

Shared Voting Power 
-0-

7.

Sole Dispositive Power 
2,368,653*

8.

Shared Dispositive Power
-0-

9.

Aggregate Amount Beneficially Owned by Each Reporting Person
2,368,653*

10.

Check if the Aggregate Amount in Row (9) Excludes Certain Shares (See Instructions)  o

11.

Percent of Class Represented by Amount in Row (9)
3.2%

12.

Type of Reporting Person (See Instructions)
OO

CUSIP No. G0450A105

1.

Names of Reporting Persons. I.R.S. Identification Nos. of above persons (entities only)
William H. Gates III

2.

Check the Appropriate Box if a Member of a Group (See Instructions)

(a)

o

(b)

o

3.

SEC Use Only

4.

Citizenship or Place of Organization
United States of America

Number of
Shares
Beneficially
Owned by
Each
Reporting
Person With

5.

Sole Voting Power
2,368,653*

6.

Shared Voting Power 
-0-

7.

Sole Dispositive Power 
2,368,653*

8.

Shared Dispositive Power
-0-

9.

Aggregate Amount Beneficially Owned by Each Reporting Person
2,368,653*

10.

Check if the Aggregate Amount in Row (9) Excludes Certain Shares (See Instructions)  o

11.

Percent of Class Represented by Amount in Row (9)
3.2%

12.

Type of Reporting Person (See Instructions)
IN

Item 1.

(a)

Name of Issuer
Arch Capital Group Ltd.

(b)

Address of Issuer’s Principal Executive Offices
Wessex House, 45 Reid Street, Hamilton HM 12, Bermuda

Item 2.

(a)

Name of Person Filing
(1)  Cascade Investment, L.L.C. (“Cascade”)

(2)  William H. Gates III

(b)

Address of Principal Business Office or, if none, Residence
(1)  2365 Carillon Point, Kirkland, Washington 98033

(2)  One Microsoft Way, Redmond, Washington 98052

(c)

Citizenship
(1)  Cascade is a limited liability company organized under the laws of the State of Washington.

(2)  William H. Gates III is a citizen of the United States of America.

(d)

Title of Class of Securities
Common Stock, $.01 par value per share

(e)

CUSIP Number
G0450A105

Item 3.

If this statement is filed pursuant to §§240.13d-1(b) or 240.13d-2(b) or (c), check whether the person filing is a:

(a)

o

Broker or dealer registered under section 15 of the Act (15 U.S.C. 78o).

(b)

o

Bank as defined in section 3(a)(6) of the Act (15 U.S.C. 78c).

(c)

o

Insurance company as defined in section 3(a)(19) of the Act (15 U.S.C. 78c).

(d)

o

Investment company registered under section 8 of the Investment Company Act of 1940 (15 U.S.C 80a-8).

(e)

o

An investment adviser in accordance with §240.13d-1(b)(1)(ii)(E);

(f)

o

An employee benefit plan or endowment fund in accordance with §240.13d-1(b)(1)(ii)(F);

(g)

o

A parent holding company or control person in accordance with § 240.13d-1(b)(1)(ii)(G);

(h)

o

A savings associations as defined in Section 3(b) of the Federal Deposit Insurance Act (12 U.S.C. 1813);

(i)

o

A church plan that is excluded from the definition of an investment company under section 3(c)(14) of the Investment Company Act of 1940 (15 U.S.C. 80a-3);

(j)

o

Group, in accordance with §240.13d-1(b)(1)(ii)(J).

Not Applicable

Item 4.

Ownership

Provide the following information regarding the aggregate number and percentage of the class of securities of the issuer identified in Item 1.

(a)

Amount beneficially owned:   

2,368,653*

(b)

Percent of class:   

3.2%

(c)

Number of shares as to which the person has:

(i)

Sole power to vote or to direct the vote   

2,368,653*

(ii)

Shared power to vote or to direct the vote    

-0-

(iii)

Sole power to dispose or to direct the disposition of   

2,368,653*

(iv)

Shared power to dispose or to direct the disposition of   

-0-

*      All shares held by Cascade may be deemed to be beneficially owned by William H. Gates III as the sole member of Cascade.

Item 5.

Ownership of Five Percent or Less of a Class

If this statement is being filed to report the fact that as of the date hereof the reporting person has ceased to be the beneficial owner of more than five percent of the class of securities, check the following   ý.

Item 6.

Ownership of More than Five Percent on Behalf of Another Person

Not Applicable

Item 7.

Identification and Classification of the Subsidiary Which Acquired the Security Being Reported on By the Parent Holding Company or Control Person

Not Applicable

Item 8.

Identification and Classification of Members of the Group

Not Applicable

Item 9.

Notice of Dissolution of Group

Not Applicable

Item 10.

Certification

By signing below I certify that, to the best of my knowledge and belief, the securities referred to above were not acquired and are not held for the purpose of or with the effect of changing or influencing the control of the issuer of the securities and were not acquired and are not held in connection with or as a participant in any transaction having that purpose or effect.

Date:  March 6, 2006

CASCADE INVESTMENT, L.L.C.

By

/s/ Michael Larson

Michael Larson, Business Manager

WILLIAM H. GATES III

/s/ Michael Larson

William H. Gates III, by Michael Larson as attorney-in-fact*

CASCADE INVESTMENT, L.L.C.

By:

/s/ Michael Larson

Michael Larson, Business Manager

WILLIAM H. GATES III

/s/ Michael Larson

William H. Gates III, by Michael Larson as attorney-in-fact*