Aaaaahh. That was arranged by Nelson Riddle and recorded for the exquisite In the Wee Small Hours album in 1955.
Sol Parker wrote the music and Hank Sanicola and Sinatra, age twenty-five, wrote the lyrics. Mark Steyn on the song's origins:
The first song he wrote was back in 1941, as boy vocalist for the Tommy Dorsey band. Sinatra and Hank Sanicola, a Dorsey staffer who used to run down Frank's numbers with him, and a song-plugger called Sol Parker cooked up a ballad called "This Love Of Mine". Young Frankie shyly showed the music and lyrics to Matt Dennis and Tom Adair (Dorsey's in-house writers, who wrote "Everything Happens To Me", "Let's Get Away From It All" and "Violets For Your Furs") and asked if they could look it over to see if it was okay. It was. It made a nice record, not just for Sinatra and Dorsey but over the years for Count Basie, Ella Fitzgerald, Ray Charles, Keely Smith, Shirley Bassey, Little Anthony and the Imperials, Lawrence Welk, Van Morrison...I'll feature another song co-written by Sinatra very soon.