BBC Radio Devon DJ David Lowe, 68, said he was unaware that a 1932 version of 'The Sun Has Got His Hat On' included the n-word.
Lowe, a broadcaster for 32 years, said it was an "innocent mistake".
He said BBC bosses declined his idea of an on-air apology, and his offer to "fall on my sword" was accepted.
The BBC said it accepted that the situation could have been handled better, and offered freelance Lowe his job back.
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The second verse of the song, recorded by the UK dance band Ambrose and his Orchestra, features the line: "He's been tanning [n-word] out in Timbuktu, now he's coming back to do the same to you." Later versions of the song omit the offensive word.
The BBC took action after a listener heard the song - part of Lowe's private record collection, broadcast on his self-produced Sunday night programme - and complained.
Lowe, a broadcaster with Radio Devon for nearly 14 years, told BBC News he had written an "unreserved apology" for use on air which he submitted to BBC managers after the song was played on April 27.
"I made a silly mistake, my first error in more than 30 years of broadcasting. I am deeply embarrassed by it," he said.
"It was a magnanimous decision of the BBC to offer my job back and if not for the impact this has had on my health I would have accepted, but my health comes first," he said.
The incident comes after calls to sack the Top Gear presenter Jeremy Clarkson were resisted by the BBC, despite unbroadcast footage showing him apparently using the n-word. He has since apologised.