A founder of the Sundance Film Festival is expected to go to prison for at least six years after he pleaded guilty in a Utah courtroom on Tuesday to molesting a young girl. The case came to light after he was recorded apologizing to a man he admitted having groped more than 25 years ago.

Sterling Van Wagenen, a director who helped establish what became the country’s most prestigious annual film exhibition, pleaded guilty in Utah County to one count of aggravated sexual abuse of a child. His lawyer, Steven Shapiro, said Mr. Van Wagenen planned to plead guilty to an additional count, involving the same girl, in neighboring Salt Lake County on Thursday. As part of a deal with prosecutors, Mr. Shapiro said he is expected to receive two sentences of six years-to-life in prison that he will serve concurrently.

Mr. Van Wagenen, 71, was charged with molesting the girl on two occasions between 2013 and 2015, when she was between 7 and 9 years old. But this was not the first allegation against him.

Sean Escobar, who as a boy was friends with two of Mr. Van Wagenen’s sons, said that Mr. Van Wagenen had touched his genitals during a sleepover at the director’s house in the early 1990s.

Mr. Van Wagenen eventually admitted to a Salt Lake County sheriff’s detective that he had touched the boy inappropriately, but the authorities dropped the case after Mr. Escobar’s parents said they did not want to press charges. (The Greater Salt Lake Unified Police Department, which absorbed the sheriff’s department, said its policy today is to move forward with child sexual abuse cases regardless of the parents’ wishes.)

Mr. Van Wagenen received a two-year “disfellowship” from the church, but went on to teach at Brigham Young University, which is closely affiliated with the church, and also directed movies for the church. Brigham Young said it had been unaware of the allegations when it hired him, and the Sundance Institute said he has had no role in the festival since 1993.

Last year, Mr. Escobar reached out to Mr. Van Wagenen, who agreed to meet with him. Mr. Van Wagenen apologized for what he had done, and said that nothing like it had happened before or since.

Mr. Escobar recorded the conversation on an iPhone he had hidden in a potted plant, then released the recording to the Truth & Transparency Foundation, an investigative website that focuses on religious reporting, thinking it might spur any other victims to come forward. Shortly afterward, the girl did.

David Sturgill, the lead prosecutor on the case in Utah County, said he was satisfied that he could resolve the case quickly and spare the victim and her family a drawn-out court process. “That’s the person I have at the top of my list of priorities when it comes, certainly, to a case like this,” he said.

Mr. Shapiro said that was his client’s wish as well. “Sterling decided that he wanted to save the family from as much difficulty as possible and enter the plea today,” he said.