[cnn-photo-caption image= http://i2.cdn.turner.com/cnn/2009/images/06/15/art.palinwave0615.gi.jpg caption="Late night TV talk show host David Letterman apologized to Alaska Gov. Sarah Palin for the second time after a recent joke by Letterman stirred controversy."]
NEW YORK (CNN) - David Letterman delivered another, more repentant apology for the off-color joke he made last week about Alaska Gov. Sarah Palin's daughter.
"It was kind of a coarse joke. There's no getting around it," Letterman said in the opening monologue of Monday night's show.
Letterman, who joked last Tuesday that Palin's "daughter was knocked up by Alex Rodriguez" at a recent Yankees game, still insisted he was referring to Palin's 18-year-old, Bristol, who gave birth to a boy in December, and not her 14-year-old, Willow.
Related video: Letterman's joke was 'perverted,' says Palin
Palin has called that a "weak, convenient excuse" for a joke that was"inappropriate" no matter which daughter was the target.
"My 14-year-old was there with me at the game. She was the only one there with me," Palin said.
Letterman said Monday night that he now realizes, "It doesn't make any difference what my intent was, it's the perception."
"As they say about jokes, if you have to explain the joke, it's not a very good joke," he said. "I take full blame for that."
"I told a bad joke. I told a joke that was beyond flawed," he said.
"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," Letterman said. "I'm sorry about it and I'll try to do better in the future.
The controversy comes at a time the late-night ratings battle between the "Late Show with David Letterman" on CBS and NBC's "The Tonight Show with Conan O'Brien" is in the spotlight.
Letterman has gained ground in the two weeks since O'Brien took over hosting duties from Jay Leno.