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The tape, featuring the unreleased song Radio Peace, was recorded on January 5 in 1970 by four 16-year-old Danish boys who succeeded in getting an interview with the couple for a school magazine.
Bids for the cassette tape, which was put up for sale along with photographs from the meeting by the former school boys, started at 100,000 crowns. The lot was valued between 200,000 and 300,000 crowns before the auction in Copenhagen on Tuesday.
It was not immediately known who bought the recording.
The recording also features Lennon and Ono humming along to Christmas songs while dancing around a Christmas tree, Lennon playing the guitar and the couple singing Give Peace a Chance and Radio Peace.
The up-tempo song, which repeats the words "this is Radio Peace," refers to a radio station of the same name that Lennon and Ono hoped to establish in Amsterdam, the auction house said.
The couple arrived in northern Denmark in late December 1969 and stayed at an isolated farm for more than a month, according to the auction house.
One of the four owners of the recording, Karsten Hojen, now 68 years old, was present at the auction. He was glad he was able to pass on the message of peace by the famous couple to the new owner.
"The meeting with John Lennon and Yoko Ono has had a great impact on our lives because we saw them as a kind of political prophets and symbols of peace," Hojen said in a statement.
"I hope the new owner will enjoy listening to our conversation and be inspired in the same way we were 50 years ago."