The cardinal, speaking at a brief news conference in Lyon, said that he would once again offer his resignation to Pope Francis. The pope had rejected the cardinal’s offer to resign after the initial conviction but had let him step down from day-to-day duties at the church while the appeal ran its course.

“This decision enables me to turn the page, and for the church in Lyon it is an opportunity to open a new chapter,” Cardinal Barbarin said. “My thoughts today — always — go to the victims.”

Matteo Bruni, a Vatican spokesman, said on Thursday that the Holy See “reiterates its proximity to all the victims of abuse” and that it was “at the side” of the Church in Lyon, which had been “sorely afflicted.”

Mr. Bruni said that the pope, “who continues to closely follow the unfolding of these painful events,” had taken note of Cardinal Barbarin’s resignation offer and would “make his decision known in good time.”

Still, the legal case against the cardinal is not over.

Lawyers for the plaintiffs announced on Thursday that they would appeal the verdict before the Cour de Cassation, the highest court in the French judiciary, which will not hold a third trial but instead verify that the law was correctly applied by lower courts.

Jean Boudot, a lawyer for one of the plaintiffs, told reporters in Lyon that the cardinal’s acquittal was “totally incoherent” because it suggested that the obligation under French law to report sexual abuse on a minor ended once victims entered adulthood, disregarding the long-lasting psychological impact of abuse.

Still, the cardinal’s acquittal did not come as a complete surprise. Prosecutors had dropped the charges against Cardinal Barbarin in 2016 after an investigation, but nine of Father Preynat’s accusers used a special procedure to force the cardinal and five other French church officials and employees of the diocese to stand trial. All five of them were acquitted in March.