The Ink Spots recorded it in 1941 although it didn't number among their hits. More on the Ink Spots here - http://en.wikipedia.org/wiki/The_Ink_Spots
An earlier version quite different version appeared before the Inkspots in 1937 by Shep Fields Rippling Rhythm (rhythm orchestra) 78rpm: Bluebird 7015 - That's When Your Heartaches Begin (Fisher-Raskin-Brown) by Shep Fields & His Rippling Rhythm Orchestra, vocal by Bob Goday. Recorded 6/3/37. Shep Fields (September 12, 1910 – February 23, 1981) was the band leader for the "Shep Fields and His Rippling Rhythm" orchestra during the Big Band era of the 1930s. More on Shep Fields here http://en.wikipedia.org/wiki/Shep_Fields
In 1953 when Elvis recorded his very early version, there was a Doo Wop version out by Billy Bun and his Buddies -below
" July 17, 1952 — Former Baptist church soloist Billy Bunn is the only male "soprano" singer in the recording field since RCA-Victor put him on the music market as an answer to cry-singer Johnny Ray. On his first record, I Need A Shoulder To Cry On, Bunn hit one octave higher than any male popular singer in the history of jazz singing." Read more about Billy Bun here http://www.vocalgroupharmony.com/ROWNEW2/ThatsWhen.htm
Elvis 1953 private recording
Billy Hill (One of the writers) - " William Joseph Hill was born in Boston Massachusetts on July 14, 1899. He studied the violin at the New England Conservatory under the tutelage of Carl Muck and attended Boston Public schools. At 17, he traveled to the west coast where he worked several jobs including as a surveyor in Death Valley and a violinist and pianist in dance halls until forming his own jazz band in Salt Lake City. Moving to New York City in 1930, Hill continued to try success in music while working another series of odd jobs. In 1933, his first hit arrived with a song called “The Last Roundup”." More on Billy Hill here - http://songwritershalloffame.org/exhibits/bio/C89
Fred Fisher (Composer) More here http://www.free-scores.com/Download-PDF-Sheet-Music-Fred-Fisher.htm