[0:00] My Bluebird Was Caught In The Rain - Ambrose & His Orchestra with Sam Browne (1930)
[7:08] I Don't Mind Walking in the Rain - Ambrose & His Orchestra (1930)
[14:14] Butterflies in the Rain - Ray Noble and His New Mayfair Dance Orchestra with Al Bowlly (1933)
[17:21] I Was Saying to the Moon - Hal Kemp & His Orchestra (1936)
[24:42] That Week in Paris - Richard Himber & His Orchestra with Stuart Allen (1938)
[27:44] Rendezvous Time In Paree - Tommy Dorsey & Jack Leonard (1939)
[31:07] If It Wasn't For The Moon - Bobby Byrne & His Orchestra with Jimmy DiPalma (1940)
[34:02] The Moon Is Back In Business - Guy Lombardo & His Royal Canadians (1948)
[36:47] Rain, Rain Go Away! - Bob Causer & His Cornellians with Dick Robertson (1932)
[40:19] Cut off :(
[40:32] I've Got Rain In My Eyes - Henry Dankers and His Star Dust Music (1937)
[47:09] The Moon Is Crying for Me - Charlie Barnet & His Orchestra with Larry Taylor (1940)
[50:34] Moon Love - Sammy Kaye
[53:43] Instrumental
[1:00:09] Rain on the Roof - Mickey Alpert & His Orchestra with Paul Small (1932)
[1:07:12] The Moon Is a Silver Dollar - Will Osborne & His Orchestra (1939)
[1:10:33] Instrumental
[1:18:14] Ukulele Moon - Guy Lombardo & His Royal Canadians (1930)
[1:21:56] Baby Boy - Joe Green Orchestra (1933)
[1:28:17] ?
[1:33:12] Comme Une Femme - Lucienne Boyer (1934)
[1:36:45] Si Petite - Lucienne Boyer (1933)
[1:40:00] Vous Qui Passez Sans Me Voir
[1:42:55] La Baignoire - Jean Sablon (1934)
[1:45:56] April in Paris - Freddy Martin & His Orchestra with Elmer Feldkamp (1934)
[1:52:18] Blue Rain - Bea Wain (1939)
Repeats from the top (needs a listen to confirm)
Special thanks to Halcyon, Hans Klopek, and J for help with this list!
Underlined songs are linked to either a YouTube video of the song or to a recording of the song from the Internet Archive. Question marks mean the song is unknown.