The song was written in 1931 by Gerald Marks and Seymour Simon and never went away.* This recording is from the 1954 album Swing Easy, the first Sinatra LP arranged by Nelson Riddle:
I like the changes this arrangement takes us through, starting off kinda sweet and then picking up a big head of steam in the second half, just perfect for the song and the singer. Frank had already sung it more than a few times by 1954, including in the 1952 film Meet Danny Wilson, and he and Nelson apparently knew exactly where they wanted to go with it.
I was going post a video of Sinatra singing this live in front of an audience of shrieking bobby-soxers in the 40s but it seems to have disappeared from YouTube, so I'll post the version from the aforementioned movie instead, which is very similar:
*Update: I was wrong. The song did go away, for about ten years. To learn why and get the great story behind this great song, see (who else?) Mark Steyn. Just a bit:
And it dawns on you that it's the perfect Sinatra song: he was cocksure and swaggering, but also the first male singer to project, seriously, vulnerability and loneliness and heartache. Hence, "All Of Me": a song for the swaggeringly vulnerable, for cocksuredness as a defense against heartache.This would make another excellent chapter in a Steyn book on Sinatra, don't you think?