Facts:

A. A.a In an agreement dated 6 October 1999, A.________ SA, in Geneva, sold to B.________ Co Ltd, in Tortola, British Virgin Islands, 20,000 tonnes of granular urea, which was to be sent to C.________ SA in Ecuador. The agreement contained an arbitration clause providing for arbitration under the Arbitration Rules of the International Chamber of Commerce (ICC) and for London as the seat of arbitration.

A dispute arose between the parties, leading the respondents to file a request for arbitration against the appellant before a sole arbitrator under the auspices of the ICC. In a first award, rendered on 18 May 2001, the sole arbitrator, dismissing the objection raised by the appellant on this point, upheld his jurisdiction over the dispute. The appellant challenged the award and the Paris Court of Appeal held the action to be inadmissible by an order issued on 11 October 2001.

On 2 May 2002, the sole arbitrator rendered a partial arbitral award in which he ordered the appellant to pay the respondents £ 12,838.53 with interest as of 18 May 2001 to cover the costs of the award on jurisdiction, and £9,927.85 with interest as of 9 December 2001 for fees covering the period of 1 May to 22 November 2001, as well as to reimburse US$ 24,000 with interest as of 4 April 2001 to cover the respondent’s down payment of fees.

A.b On 24 September 2002, the respondents requested that the Geneva Bureau of Court Cases notify the appellant of the order to pay in case no. ... (hereafter: Action A), the equivalent in Swiss francs of the amounts awarded in the partial arbitral award of 2 May 2002. This payment order was challenged.