The episode featured a profile of Kahlo's artist husband Diego Rivera. In it, she reads from her essay Portrait of Diego.
"With an Asiatic head from which dark hair grows, so thin and fine that it appears to float in the air, he is a enormous, huge child, with a friendly face and a sad gaze", what is thought to be Kahlo's voice reads liltingly, in Spanish. The recording is thought to have been made in 1953 or 1954.
The library's role is to preserve and make public documentary sound archives for Mexico.
"Frida's voice has always been a great enigma, a never-ending search", Granados told a press conference.
Frida Kahlo's celebrated art and iconic unibrowed face are globally recognizable, but her voice was thought to have died with her - until now.
The radio show the recording was taken from, El Bachiller, was named for its host Álvaro Gálvez y Fuentes "El Bachiller", who has been described by the ministry of culture as "the main documentalist of the voices of his time".
"Have you ever imagined how the voice of #FridaKahlo could have sounded?" the sound library, Fonoteca Nacional, asked in a Tweet.