Harry Nilsson joined Carly Simon in London in November of 1972 to record the backing vocals for "Ballad of a Vain Man as "You're So Vain" was called until it was renamed as the record was being mastered.
Richard Perry produced the recording.
She replied, "I guess it was kind of chance in a way. I was in London, it was 1973, 1972 I guess, and he happened to call at the studio while I was doing the background vocals with Harry Nilsson. Mick said 'Hey, what cha doin'?' and I said 'We're doing some backup vocals on a song of mine....why don't you come down and sing with us?' [...] So Mick and Harry and I stood around the mike singing 'you're so vain' and Harry was such a gentleman - he knew the chemistry was between me and Mick; in terms of the singing, so he sort of bowed out saying 'The two of you have a real blend - you should do it yourselves.' And that's how it happened."
-  Stephen Davis . Gotham Books (2012) "More Room in a Broken Heart: The True Adventures of Carly Simon"