Allmusic.com gives this information:

The sister duo Patience and Prudence reached the Top Ten in 1956 with a fluke hit, the ghostly TONIGHT YOU BELONG TO ME. The daughters of a professional musician — Mark McIntyre, who accompanied Frank Sinatra on piano during the mid-'40s — Patience and Prudence McIntyre reportedly learned the song while at summer camp in 1955 (when they were nine and 12, respectively). Their father recognized it as a Billy Rose hit from several decades earlier, and was impressed enough to work up the arrangement as a demo intended for cabaret singer Lisa Kirk (wife of Robert Wells). McIntyre also recorded a version with Patience and Prudence's vocals, strictly for personal reasons, but gave a dub to studio friend Ross Bagdasarian, who passed the record on to Si Waronker of Liberty Records.Waronker loved the song, so Patience and Prudence signed to Liberty and recut the song with overdubs to strengthen their ethereal vocals, then watched as the record soared to number four nationally (with several cover versions eventually coming along). After performing on Perry Como's television show, they chose for a follow-up single GONNA GET ALONG WITHOUT YOU NOW (earlier a minor hit for Teresa Brewer), which just missed the Top Ten later in 1956. The duo recorded several more singles — including a pairing with fellow Liberty act Mike Clifford and several singles for the Chattahoochee label — but never reached the charts again. They reunited for a 1978 performance on a Dick Clark feature, and earned a surprisingly long lease on life after their songs were used in a pair of films, 1999's Election (GONNA GET ALONG WITHOUT YOU NOW) and 2001's Ghost World (A SMILE AND A RIBBON).

Indiana University Sheet Music Collections gives this:

Title: TONIGHT YOU BELONG TO ME
Composer: David, Lee
Lyricist: Rose, Billy, 1899-1966
Publisher: Harry Waterson Inc.
Place of publication: New York
Date of publication: 1926
First Line: Once more we meet, you look so sweet, dear,
Chorus First Line: Though you belong to somebody else, tonight you belong to me

The Online 78 rpm Discographical Project lists these recordings: (Note that it had a revival in the late 30s as a western swing tune.)

Teddy Phillips, (No Date Given)
Adrian Schubert's Orchestra, 1926
Alma Rotter, 1926
Arthur Fields, 1926
Baltimore Salon Orchestra, 1926
Blue Ribbon Quartet, 1926
Bob Haring & His Orchestra, 1926
Broadway Dance Orchestra, c1926
California Melody Men, 1926
Lee Morse, 1926
Marlborough Dance Orchestra, 1926
McAlpin Dance Orchestra, 1926
Nathan Glantz & His Orchestra, 1926
Paramount Dance Orchestra, c1926
Ponce Sisters, 1926
Roger Wolfe Kahn & His Orchestra with vocal by Franklin Baur, 1926
Schubert's Society Orchestra, 1926
William Robyn, 1926
Harmony Dance Orchestra, c1927
Cliff Bruner's Texas Wanderers, 1937
Hank Penny's Radio Cowboys, 1939
Johnny Fenton & His Orchestra, 1939
The Tennessee Ramblers, 1939
Eddy Howard, 1940
Erskine Hawkins & His Orchestra, 1940
Peggy Lee, 1940s
Russ Case, 1940s
Gene Austin, 1941
Joe Costa; Orchestra; Joe Lipman, c1950
Frankie Laine, 1952