28 U.S.C § 2101 (stay of execution pending application for writ of certiorari)

...

(f) In any case in which the final judgment or decree of any court is subject to review by the Supreme Court on writ of certiorari, the execution and enforcement of such judgment or decree may be stayed for a reasonable time to enable the party aggrieved to obtain a writ of certiorari from the Supreme Court. The stay may be granted by a judge of the court rendering the judgment or decree or by a justice of the Supreme Court, and may be conditioned on the giving of security, approved by such judge or justice, that if the aggrieved party fails to make application for such writ within the period allotted therefor, or fails to obtain an order granting his application, or fails to make his plea good in the Supreme Court, he shall answer for all damages and costs which the other party may sustain by reason of the stay.