Sir Terry Wogan's Radio 2 show breached broadcasting guidelines, the BBC Trust has said, after the veteran presenter appeared to make light of the Costa Concordia tragedy.
The radio star made the comments, which listeners complained were "inappropriate and offensive", nine days after the cruise ship ran aground in January.
At the time, it was reported that at least 11 people had died - the number later rose to 32 - and the search for survivors was still ongoing.
Sir Terry made a joke after disco track Rock The Boat was played on his two-hour live Sunday morning show, Weekend Wogan.
Operations to remove fuel from the Costa Concordia a month after it ran aground
As the song faded, he mused on whether it had been an appropriate song and joked about the ship's captain and wanting to be the first in the lifeboat if the BBC went down.
"Frankly if I had my time over again, and given the boating tragedy in Italy, I mightn't have picked that as an opening song," he chuckled.
"Rock The Boat, argh, Captain Coward," he added.
Later, after the news bulletin he said to the announcer: "I don't know about you ... but I'll be the last to leave the BBC."
He added later: "Not sinking is it? Me first, never mind the women and children, I'm not even Italian."
The comments were referred to the BBC Trust after the Editorial Complaints Unit ruled that they did not warrant a public apology, even though they had been "inappropriate" and "ran the risk of causing offence".
The trust's Editorial Standards Committee, which acts as the final arbiter of appeals if complainants are unhappy with the way their initial complaints have been dealt with by BBC management, said it was surprised that there had not been an on-air apology.
Sir Terry's remarks were "characteristically self-deprecating, joking about his own lack of bravery rather than the victims of the tragedy itself," The trust said.
"In this context the committee did not believe there had been any intention to cause offence.
"The committee, however, did conclude that there was a real risk of causing offence and in this context the guidelines had been breached.
"The committee expressed surprise that the BBC did not apologise on-air on the day," it said.
But it added that as the BBC had accepted that the remarks and music were inappropriate and had apologised to the listener, there should be no further sanction.
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