The married couple since July 2004 welcomed a baby boy, Kal-el Coppola Cage today in NYC. Here's the background on the babies name from People:
As it turns out, Kal-el is the comic-book birth name of Superman on the planet Krypton. The name Coppola, however, is Cage's family name. His uncle is the director Francis Ford Coppola.
Image Source: Cage Factor