Magistrates in Lyon found that Barbarin had an obligation to report the Rev. Bernard Preynat to civil authorities and gave the cardinal a six-month suspended prison sentence. Barbarin offered to resign.
Preynat, who is scheduled to be tried on sexual violence charges next year, has confessed to abusing Boy Scouts in the 1970s and 1980s. People who said they were among the victims accused Barbarin and other church officials of covering up the priest's crimes for years.
Nine victims brought the case to trial. A group of Preynat's victims hailed the unanticipated conviction as a victory for child protection and a strong signal that church leaders will be held accountable.
"We see that no one is above the law. We have been heard by the court," said Francois Devaux, president of La Parole Liberee (Lift the Burden of Silence.)
The verdict came as a surprise since the statute of limitations had expired on some charges. At the end of the trial, prosecutors argued for an acquittal, saying there were no grounds to prove legal wrongdoing.
Five co-defendants - an archbishop, a bishop, a priest and two other officials - were acquitted.
Barbarin's conviction furthers the crisis facing the Catholic Church's embattled hierarchy. The sex abuse scandal recently led to Australian Cardinal George Pell's abuse conviction and the defrocking of America's ex-Cardinal Theodore McCarrick for sexually molested minors and adult men, including during confession..
The scandal swept up Pope Francis last year after he defended a bishop accused of veering up abuse in Chile. Francis is facing new questions about his backing for a bishop now under investigation for sexual abuse in his native Argentina.
With the confidence of Catholic faithful in the church hierarchy shaken, Francis brought church leaders from around the world to a summit last month at which he vowed an "all-out battle" to end abuse but issued no new rules to stop cover-ups.
In the French court's decision, read by The Associated Press, the magistrates wrote that Barbarin "had the obligation to report" accusations against Preynat between July 2014 and June 2015 because the priest's accusers didn't request ecclesiastic secrecy.
The victims' allegation of a cover-up that allowed Preynat to be in contact with children until his 2015 retirement was thrown out of court in 2016 for insufficient evidence.
Alexandre Hezez, one of those who put it back on the docket through a direct approach French law allows as a recourse, met with Barbarin in November 2014 and kept informing him there were probably other Preynat victims, according to the court ruling.
Barbarin "didn't take the initiative despite Hezez's request and insistence," the magistrates wrote.
The cardinal was not in court when the decision was handed down. His lawyer, Jean-Felix Luciani, said he will appeal.
"This is a decision that is not fair at the juridical level," Luciani said. He added: "We hope that at the next step, justice will be done."
Barbarin, 68, nevertheless said he would offer to resign when he meets with Pope Francis "in a few days."
"I have decided to go and see the Holy Father to offer him my resignation," he said in a brief statement, expressing his compassion for Preynat's victims.
Barbarin was made a cardinal by St. John Paul II in 2003; a year after John Paul made him archbishop of Lyon.
Unlike most cardinals, he does not have a large presence outside his home country. He only serves as a member of a single Vatican office, the congregation for religious orders. Even cardinals who live much farther from Rome work on two or three congregations at a time.
The Vatican didn't immediately respond to a request for comment.
Pope Francis previously said the French justice system should take its course, but praised Barbarin as "brave."
Francis now has to weigh whether to accept, reject or delay Barbarin's offer of resignation.
Last year, Francis reluctantly accepted the resignation of one of his key supporters, Washington Cardinal Donald Wuerl, after he was implicated in cover-ups from decades ago.
The cases were revealed by a Pennsylvania grand jury report. Francis kept Wuerl on as a temporary administrator pending the appointment of a replacement.
He also accepted the resignation last year of Adelaide, Australia Archbishop Philip Wilson after he was convicted of covering up abuse. The conviction was reversed on appeal, but Wilson already had been replaced. His future status is unclear.
In France, the priest's said top clergy had been aware of Preynat's actions since 1991, but allowed him to be in contact with children until his 2015 retirement.
Several men recounted during the four-day trial in January the fear and shame they felt after they were abused.
Christian Burdet, 53, testified that Preynat forced him to go into his tent when he was a 10-year-old Scout.
Describing years of suffering, Burdet said he wanted to "understand how this system was put in place" and help other victims to speak out.
A date for Preynat's trial next year has not been set. Only 13 cases will be heard in court, as the statute of limitations had expired for others. The priest is estimated to have abused 85 victims.
Devaux, of the victims' advocacy group, said the cardinal's conviction sent strong signals to both victims and the top ranks of the Catholic Church.
Top church officials need to "re-evaluate their sacred dimension. We're on Earth here, and while some things are acceptable, others are not."
A lawyer for some of Preynat's victims, Yves Sauvayre, called the verdict "historic."
"The cardinal is convicted because he didn't do what needed to be done," he said.
Anne Barrett Doyle, of the online resource BishopAccountability.org, said Thursday's decision was "a significant step forward" in making the Catholic hierarchy answerable for conduct it hid for decades.
"In affirming the victims' claims, the French court implicitly rejected the grab bag of excuses that Barbarin and other Catholic bishops use to justify non-reporting," she said in a statement.
The decision against Barbarin was handed down less than two weeks after the conviction of another "prince" of the church, Pell, was announced in his native Australia of sexually abusing two youths. He too is appealing.