What song is being sung in the bar while George is praying?

When George Bailey first enters Martini's bar in It's a Wonderful Life and prays for help, there is a song being sung in the background that isn't in English (possibly Italian?), and while I can't understand the lyrics, the tone doesn't seem like a Christmas song.

Can anyone identify the song, and does it have any connection to Christmas or to the message of the movie?

It is identified in the Youtube comments. The song is "Vieni Vieni", sung by Adriana Caselotti, (the voice of Snow White.) Here's another version on Youtube sung by Tino Rossi.