Exhibit (a)(7)(B)

OFFERING CIRCULAR SUPPLEMENT

Arch Coal, Inc.

Offer to Pay a Premium Upon the Conversion

of up to an Aggregate of 2,874,926 Shares of Its

5% Perpetual Cumulative Convertible Preferred Stock

(Liquidation Preference $50.00 Per Share) to Common Stock

CUSIP No. 039380 20 9

ISIN No. US0393802097

      This offering circular supplement relates to the offer by Arch Coal, Inc., a Delaware corporation (the “Company”), to pay a premium to holders of any and all of its outstanding 5% Perpetual Cumulative Convertible Preferred Stock (Liquidation Preference $50.00 Per Share) (“Preferred Stock”) who elect to convert their shares of Preferred Stock to the Company’s Common Stock, $.01 par value (“Common Stock”), in accordance with the terms of the Preferred Stock and upon the terms and subject to the conditions set forth in the Company’s offering circular, dated November 30, 2005 (the “Offering Circular”), and in the accompanying letter of transmittal. This offering circular supplement updates the Offering Circular and should be read in conjunction with the Offering Circular.

      The Company has determined to change the formula set forth in the Offering Circular for calculating the number of shares that will be paid as a premium in the conversion offer for each share of Preferred Stock validly tendered and accepted for conversion. As changed, the premium offered in the conversion offer is an amount of shares of our Common Stock valued at $3.50, as determined by dividing (i) $3.50 by (ii) the volume-weighted average of the reported sales prices on the New York Stock Exchange of our Common Stock during the five trading days ending at the close of the second trading day prior to the expiration of the conversion offer (including any extension), per share of Preferred Stock validly tendered and accepted for conversion. Other than this change to the formula used to determine the number of shares of Common Stock that you will receive as a premium if you participate in the conversion offer, the terms and conditions set forth in the Offering Circular remain in effect, including the expiration of the conversion offer at 12:00 midnight, New York City time, on December 29, 2005, unless extended or earlier terminated.

      Enclosed is a new letter of transmittal for your Preferred Stock. If you have not already returned a completed letter of transmittal with respect to your Preferred Stock, please complete the enclosed letter of transmittal if you wish to participate in the conversion offer. If you have already returned a completed letter of transmittal, you do not need to return the enclosed letter of transmittal unless you wish to change the information set forth in the letter of transmittal that you have already returned. This new letter of transmittal, when received, will supersede your earlier letter of transmittal. In any event, you may validly withdraw shares of Preferred Stock that you have already tendered or that you may tender at any time prior to the expiration date of the conversion offer. In addition, if not previously returned, you may withdraw any shares of Preferred Stock that you tender that are not accepted by us for conversion before January 30, 2006, which is 40 business days from the commencement of the conversion offer. For a withdrawal of shares of Preferred Stock to be effective, you must comply with the appropriate procedures of DTC’s ATOP system prior to the expiration date or, if not accepted by us before January 30, 2006, the 40th business day after the commencement of the conversion offer. Any notice of withdrawal must identify the shares of Preferred Stock to be withdrawn, including the name and number of the account at DTC to be credited and otherwise comply with the procedures of DTC.

      If you have questions regarding the procedures for tendering in the conversion offer or require assistance in tendering your shares of Preferred Stock, please contact American Stock Transfer & Trust Company, the information agent for the conversion offer, at (718) 921-8317 or toll-free at (877) 248-6417. If you would like additional copies of this offering circular supplement, the Offering Circular, our annual, quarterly and current reports, proxy statement and other information that we incorporate by reference in the Offering Circular, please contact either the information agent at either telephone number set forth above or Investor Relations at Arch Coal at (314) 994-2700. Holders of Preferred Stock may also contact their brokers, dealers, commercial banks, trust companies or other nominees through which they hold their Preferred Stock with questions and requests for assistance.

The date of this offering circular supplement is December 5, 2005.