Last Call with Carson Daly is listing The Cure for Friday night, and the episode is new, not a repeat. If that listing is accurate, then there's a very good chance that they're airing one of the other songs they played at the Dec. 12th taping - Sleep When I'm Dead or Boys Don't Cry.
Note: I thought this would be pretty clear. The Cure are NOT in L.A. for a new taping. They played 3 songs at the Dec. 12th taping, and it would just be one of the unaired songs that are listed above.
Update: The official site confirms that it will be Sleep When I'm Dead. (Thanks Jason)