Comedian Russell Kane was asked to explain why there is nothing funny about the fact that the Queen "must have had sex at least four times" on the programme, which aired at 6.30 p.m. GMT (1.30 p.m. ET) on 21 April.
Kane's observations included the line: "Four times we have to think of republicanism as we imagine four children emerging from Her Majesty's vulva."
It follows a similar ruling from the BBC's governing body, the BBC Trust. It said panellists' comments were "personal, intrusive, and demeaning."
The BBC said the "Don't Make Me Laugh" broadcast was a "regrettable failure of editorial judgment and compliance processes." The jokes were not flagged properly before transmission - even after the BBC scheduled the show on the Queen's birthday.
The BBC cancelled a repeat of the David Baddiel-presented programme, removed it from iPlayer, and moved the rest of the series to 11 p.m. The broadcaster also apologised to Buckingham Palace. A version of this was published on the BBC's website.
The BBC has since confirmed that "Don't Make Me Laugh" will not return for another series. A spokeswoman told The Guardian: "We're lucky to receive hundreds of great ideas from brilliant comedians who want to work with Radio 4, and we always bring a mix of returning shows to our audiences whilst also finding space for new programmes in our packed schedule."