In a release from the show's production company, Letterman said he referred the matter to the Manhattan district attorney's office and that an investigation ended in an arrest Thursday. Letterman did not identify the person he said was arrested.
As part of the investigation, Letterman said he issued a "phony" $2 million check to the individual and the arrest followed - along with testimony by Letterman.
"This morning, I did something I've never done in my life," said Letterman. "I had to go downtown and testify before a grand jury."
In his testimony, he said he acknowledged sexual relationships with members of his staff. It was not immediately clear when the relationships took place; Letterman and longtime girlfriend Regina Lasko married in March. The couple began dating in 1986 and have a son, Harry, born in November 2003.
"My response to that is, yes I have," Letterman said. "Would it be embarrassing if it were made public? Perhaps it would. I feel like I need to protect these people. I need to certainly protect my family."
CBS spokesman Chris Ender said Thursday that "Letterman's comments on the broadcast tonight speak for themselves."
It's the second set of embarrassing headlines for Letterman in four months. In June, he apologized to Sarah Palin for making a crude joke about the former Republican vice presidential candidate's 14-year-old daughter. Although there was a small "fire Letterman" demonstration outside of his studio later, CBS stood by its late-night star.
After nearly 15 years in second place to NBC's Jay Leno in the ratings, Letterman took over the top spot this summer after Conan O'Brien became "Tonight" show host.
Letterman's CBS "Late Show" has been on the air since 1993 and before that, he had a late-night show on NBC.
Alicia Maxey Greene, a spokeswoman for the Manhattan District Attorney's office, declined to comment.
AP Television Writer Lynn Elber in Los Angeles and Associated Press Writer Tom McElroy in New York contributed to this report.
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