And the shameless grovelling paid off when the 47-year-old singer became only the second artist ever allowed to use their work.
The song, which features on new single HUNG UP, sees Madonna join the Fugees as the only other musicians to be given the Abba seal of approval.
The group used part of THE NAME OF THE GAME on their 1996 track RUMBLE IN THE JUNGLE.
Madonna says, "I had to send my emissary to Stockholm with a letter begging them and telling them how much I love their music.
"They never let anyone sample their music. Thank God they didn't say no."
Andersson says, "We admire Madonna so much and always have done.
"She has got guts and has been around for 21 years. That is not bad going.
"Hung Up is really good. If it wasn't any good we would not have said yes. It is a wonderful track - 100 per cent solid pop music.''