It was the television equivalent of sticking an ex with the tab.

Nearly a month after the acrimonious exit of Stephen Colbert from his late-night show, CBS agreed on Tuesday to bear the cost of one of the host’s stunts during his sign-off: playing a “Peanuts” tune to which the network did not own the rights.

This was no accident.

Mr. Colbert had been discussing how the copyright holders of the “Peanuts” music catalog had recently sued the federal government and three companies over their unauthorized use of songs when his house band, led by Louis Cato, began playing the “Linus and Lucy” theme during the final episode of “The Late Show With Stephen Colbert.”

“Anyone illegally using that music is going to have to pay through the nose,” Mr. Colbert said on the show, a saxophone riff of the song filling the Ed Sullivan Theater.

As Mr. Colbert feigned astonishment over the choice of music, Mr. Cato, the bandleader, put his finger to mouth pensively.

“Oh no. I hope this doesn’t cost CBS any money,” Mr. Colbert said.

In a statement on Tuesday, Lee Mendelson Film Productions, which owns the copyright of the “Peanuts” music composed by Vince Guaraldi, said CBS had agreed to pay an undisclosed amount for using the song during Mr. Colbert’s final show on May 21.

The proceeds will be donated by the production company to World Central Kitchen, a relief group founded by the celebrity chef José Andrés, to whom Mr. Colbert had presented a check for a donation of nearly $2.5 million on his second-to-last show.

Jason Mendelson, the chairman of Lee Mendelson Film Productions, described the stunt as funny and entertaining.

“A principal goal of our enforcement actions is to educate individuals, businesses and government entities about the need to obtain written license agreements to use music in a commercial setting,” Mr. Mendelson said in the company’s statement.

CBS referred to the same news release when asked for comment on Tuesday.

The network did not say whether it would seek compensation from Mr. Colbert, whose show was canceled last year by CBS in what it said was a financial decision.

To Mr. Colbert and his supporters, the decision to end his run at CBS, which began in 2015, and lowered the curtain on the more than three-decade-old “Late Show” franchise, was a curious one that they suggested was connected to politics.

Many speculated that his exit was prompted by his relentless mockery of President Trump, whose administration cleared the way last week for a $111 billion merger of Paramount Skydance, the parent of CBS, with Warner Bros. Discovery.

A representative for Mr. Colbert did not respond to a request for comment about CBS having to pay for the unauthorized use of the “Linus and Lucy” tune.