Update, Tuesday morning:

Here's Gov. Palin's response to Letterman apology, posted early this morning on her Facebook page:

"Of course it's accepted on behalf of young women, like my daughters, who hope men who 'joke' about public displays of sexual exploitation of girls will soon evolve.

"Letterman certainly has the right to 'joke' about whatever he wants to, and thankfully we have the right to express our reaction. And this is all thanks to our U.S. Military women and men putting their lives on the line for us to secure America's Right to Free Speech - in this case, may that right be used to promote equality and respect."

Original post from Monday afternoon:

From the New York Times:

David Letterman directly apologized to Gov. Sarah Palin and her daughters on his program Monday night, saying he took responsibility for a joke that had offended Ms. Palin, her family, and her supporters.

Mr. Letterman opened the desk portion of his show with the apology in which he said he wanted to say he was sorry to “to the two daughters involved, Bristol and Willow, and also to the governor and her family and everybody else who was outraged by the joke.” Two weeks ago on his “Late Show” program on CBS, he had joked about Governor Palin attending a Yankee game with her daughter.

From the L.A. Times: Letterman apologizes for joke, but Palin supporters vow to continue fight

Here are Letterman's full remarks from tonight's show, as reported by Newsday:

“All right, here – I’ve been thinking about this situation with Governor Palin and her family now for about a week – it was a week ago tonight, and maybe you know about it, maybe you don’t know about it. But there was a joke that I told, and I thought I was telling it about the older daughter being at Yankee Stadium. And it was kind of a coarse joke. There’s no getting around it, but I never thought it was anybody other than the older daughter, and before the show, I checked to make sure in fact that she is of legal age, 18. Yeah. But the joke really, in and of itself, can’t be defended. The next day, people are outraged. They’re angry at me because they said, ‘How could you make a lousy joke like that about the 14-year-old girl who was at the ball game?’ And I had, honestly, no idea that the 14-year-old girl, I had no idea that anybody was at the ball game except the Governor and I was told at the time she was there with Rudy Giuliani…And I really should have made the joke about Rudy…” (audience applauds) “But I didn’t, and now people are getting angry and they’re saying, ‘Well, how can you say something like that about a 14-year-old girl, and does that make you feel good to make those horrible jokes about a kid who’s completely innocent, minding her own business,’ and, turns out, she was at the ball game. I had no idea she was there. So she’s now at the ball game and people think that I made the joke about her. And, but still, 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, and you can’t really be doing jokes like that. And I understand, of course, why people are upset. I would be upset myself.

“And then I was watching the Jim Lehrer ‘Newshour’ – this commentator, the columnist Mark Shields, was talking about how I had made this indefensible joke about the 14-year-old girl, and I thought, ‘Oh, boy, now I’m beginning to understand what the problem is here. It’s the perception rather than the intent.’ It doesn’t make any difference what my intent was, it’s the perception. And, as they say about jokes, if you have to explain the joke, it’s not a very good joke. And I’m certainly – ” (audience applause) “– thank you. Well, my responsibility – I take full blame for that. I told a bad joke. I told a joke that was beyond flawed, and my intent is completely meaningless compared to the perception. And since it was a joke I told, I feel that I need to do the right thing here and apologize for having told that joke. It’s not your fault that it was misunderstood, it’s my fault. That it was misunderstood.” (audience applauds) “Thank you. So I would like to apologize, especially to the two daughters involved, Bristol and Willow, and also to the Governor and her family and everybody else who was outraged by the joke. I’m sorry about it and I’ll try to do better in the future. Thank you very much.” (audience applause)