It's by Marvin Rainwater, who also recorded the definitive version in 1957. There was a great whistling break on it, sounded almost like yodeling. One's memory is imprecise (and incomplete), however...

Gonna buy me a bluebird
Let him sing me a song
'Cause my heart's been lonely
Much too long

Gonna paint me a rainbow
In a heaven of blue
'Cause my heart's been crying
O-ver you
(Surely there's more...)