Mr. LaRavia said the skit involved “crawling around the floor looking for cigar butts,” and that it “certainly got a big reaction out of the audience.”

Gina Maiola, a spokeswoman for Ms. Ivey, said the interview was discovered as Auburn was digitizing years of old audio tape in its archives. The state school’s government affairs office made the governor’s office aware of the recording on Tuesday, and Ms. Ivey first listened to it on Wednesday, Ms. Maiola said. The audio was released by the governor’s office on Thursday.

In a prepared statement, Ms. Ivey said that she did not recall the skit or the interview, but that she would “not deny what is the obvious.”

“I fully acknowledge — with genuine remorse — my participation in a skit like that back when I was a senior in college,” she said. “While some may attempt to excuse this as acceptable behavior for a college student during the mid-1960s, that is not who I am today, and it is not what my administration represents all these years later.”

She added: “I offer my heartfelt apologies for the pain and embarrassment this causes, and I will do all I can — going forward — to help show the nation that the Alabama of today is a far cry from the Alabama of the 1960s. We have come a long way, for sure, but we still have a long way to go.”

Alabama must now deal with the fact that a second governor in a row has been tainted by high-profile scandal. In April 2017, Ms. Ivey, then the lieutenant governor, took over as governor after her Republican predecessor, Robert Bentley, resigned amid a sex scandal and an impeachment inquiry.

Ms. Ivey, who had previously served as state treasurer, became the second woman to lead the state.

In 2017, she signed into law the Alabama Memorial Preservation Act, which prohibits moving or changing memorials, including the numerous memorials to the Confederacy that dot the state.