“Put the Blame on Mame” is a song by Allan Roberts and Doris Fisher, originally written for the classic film noir Gilda (1946) in which it was sung by the title character, played by Rita Hayworth with the singing voice of Anita Kert Ellis dubbed in, while Glen Ford looked and fumed.

Anita Kert Ellis has a Wikipedia page in which she is named Anita Ellis, nee Kert, that describes the behind the scenes of Gilda:

For her work in Gilda, her role in the film was by greatly underplayed and even kept secret by the producers who wanted fans to believe Hayworth was the singer. They went as far as to put Rita Hayworth’s name in the credit of the soundtrack instead of Anita Ellis. Hayworth also bore some resentment towards the studio for not allowing her to sing her own parts and the embarrassment it caused when she was asked to sing by fans who thought she was the voice of Gilda. There were claims made that Hayworth had sung the acoustic guitar version  of ‘Put the Blame on Mame’ but this was untrue. Ellis dubbed all of the vocal parts since both their voices could not be used, Ellis’s singing voice was too unlike Hayworth’s for Hayworth’s voice to be used in different parts of the film.