Barbarin had travelled to the Vatican on Monday to present his resignation following the 7th March verdict and six-month suspended sentence.
Vatican spokesman Alessandro Gisotti said that during the audience, Francis did not accept the resignation and instead asked Barbarin to do what he thinks is best for the archdiocese.
Barbarin has decided to take time away and has asked his deputy in Lyon, the Rev Yves Baumgarten, to assume leadership of the archdiocese, Mr Gisotti said.
"The Holy See repeats its closeness to victims of abuse, to the faithful of the archdiocese of Lyon and the French church who are living in a particularly difficult moment," Mr Gisotti said.
The French court found that Barbarin had an obligation to report the Rev Bernard Preynat to civil authorities when he learned of his abuse.
Preynat, who is scheduled to be tried on sexual violence charges next year, confessed to abusing Boy Scouts in the 1970s and 1980s. His victims accuse Barbarin and other church authorities of covering up for him for years.
Barbarin has appealed against the conviction, which could have influenced Francis's decision to not immediately accept his resignation. Francis has been reluctant to take such measures in case they influence the outcome of trials.
Last year, he 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. Wuerl was never criminally charged, but he determined he could no longer effectively lead the archdiocese after he lost the trust of the flock and some of his priests.
Francis also accepted the resignation last year of Philip Wilson, archbishop of Adelaide in Australia, after he was convicted of covering up abuse.
Wilson's conviction was reversed on appeal but he had already had been replaced. His future status is unclear.
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