In which of Shakespeare's plays did Sir John Falstaff die?

In Shakespeare's Henry V, Falstaff is seriously ill and his friends say the old knight is dying of a broken heart because King Henry has rejected him. At one point, Mistress Quickly declares, "The king has killed his heart" (2.1.86), and everyone pretty much agrees.

Soon after the statement by Quickly, everyone is told flatly that Falstaff "is dead" (2.3.5 Henry V). Apparently, he's succumbed to some kind of nasty venereal disease (STD), which is sad but also fitting in light of Falstaff's penchant for hanging out in brothels and taverns.