The veteran US presenter David Letterman has been forced to make an abject on-air apology after a joke he told at the expense of Sarah Palin's daughter spectacularly backfired.

"It was a coarse joke, a bad joke," Letterman confessed on last night's edition of CBS's Late Show. "The joke, really, in and of itself, can't be defended."

The trouble started a week ago when the presenter made a string of gags on Monday's edition about the supposed highlights of a trip to New York by Mrs Palin and her family.

He claimed that the Governor of Alaska "bought make-up at Bloomingdales's to update her slutty flight attendant look" - a swipe at Mrs Palin's supposed fashion mistakes as Republican candidate for Vice-President last year.

But the crack that brought an avalanche of protests down on his head was when he referred to a family trip to a baseball game.

He said that an awkward moment occurred during the seventh innings when Mrs Palin's daughter was "knocked up by Alex Rodriguez", the third baseman for the Yankees team.

Letterman maintains that he was under the impression that the daughter who went to the game with Mr and Mrs Palin was Bristol, an 18-year-old unmarried mother. In fact, it was Willow, 14.

Todd Palin issued a furious statement saying that "any jokes about raping my 14-year-old are despicable", and Mrs Palin referred to the furore as "sexually perverted comments made by a 62-year-old male celebrity".

Since then Letterman has tried to skate over the row. In Wednesday's Late Show he conceded "we make mistakes left and right", and said he would never make jokes about raping or having sex with a 14-year-old girl as: "I don't think it's funny."

But he continued to lard his speech with more gags, inviting the Palins on his show. "Or leave Todd at home," he suggested.

This performance failed to mollify either the Palins or their supporters, so on Thursday's show he returned to the row, joking that everything was fine between him and blood-sports enthusiast Mrs Palin because she had called and offered to take him hunting.

On Friday Mrs Palin hit back in an appearance on the rival NBC network's Today show, taking the moral high ground.

"I would like to see him apologise to young women across the country for contributing to that kind of thread that is throughout our culture, that makes it sound like it is OK to talk about young girls in that way, where it's kind of OK, accepted and funny to talk about statutory rape," she said severely. "It's not cool. It's not funny.

"No wonder young girls especially have such low self-esteem in America when we think it's funny for a so-called comedian to get away with such a remark as he did."

Last night Letterman bowed to pressure and finally issued an apology to both daughters "and also to the Governor and her family and everybody else who was outraged by the joke" - although he continued to justify himself.

"I never thought it was (about) anybody other than the older daughter, and before the show I checked to make sure, in fact, that she is of legal age, 18," he said.

"I'm wondering, well, what can I do to help people understand that I would never make a joke like this? I've never made jokes like this, as long as we've been on the air, 30 long years...

"It's not your fault it was misunderstood, it's my fault it was misunderstood."

Publicity about the row has benefited both sides. The furore has helped to keep Mrs Palin's profile high, at a time when her possible run for President in three years' time appears to have bogged down in family feuding over Bristol's baby, while other female torchbearers for the right threaten to overtake her.

The row appears also to have boosted Letterman while he is competing for ratings with his rival Conan O'Brien on NBC's Tonight.

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