The move follows a period in which Francis faced mounting pressure to take action against Wilson, a top figure in an Australian church that has been deeply scarred by decades of abuse cases.
It is the second time in three days in which Francis has accepted a resignation stemming from sexual abuse - part of a global reckoning for a church that has long been reluctant to discipline those in its highest echelons.
On Saturday, the former archbishop of Washington, Theodore McCarrick, became the first cardinal in history to step down due to allegations of sexual abuse.
Wilson had previously refused to resign, saying he was entitled to due process and was pressing forward with an appeal.
He was sentenced recently to a 12-month detention sentence for what an Australian court says was his failure to report cases of sexual abuse carried out in the 1970s by a priest, James Fletcher.
Two altar boys said they told Wilson about abuse at the hands of Fletcher, and in the courtroom, one described how the priest had forced him to strip and kneel as he masturbated.
Prosecutors said Wilson did nothing with the information.
Wilson denies that he was told about the abuse. Wilson, who has been diagnosed with Alzheimer’s disease, stepped away from his day-to-day work as Archbishop of Adelaide after being convicted. But he retained the title of archbishop.
“While the judicial process will continue, Archbishop Wilson’s resignation is the next chapter in a heartbreaking story of people who were sexually abused at the hands of Jim Fletcher, and whose lives were forever changed,” the Australian Catholic Bishops Conference president, Archbishop Mark Coleridge, said in a statement.
“This decision may bring some comfort to them, despite the ongoing pain they bear.”
In 2004, Fletcher was found guilty on nine counts of sexual abuse. He died in prison in 2006.
In recent weeks, a series of political leaders in Australia, including Prime Minister Malcolm Turnbull, had called on Francis to take action, with Turnbull saying: “The time has come for the pope to sack him.”
For Francis, the latest resignation underscores a new wave of challenges he is facing about how to handle not just the priests who abused children, but also the cardinals and bishops who allowed the abuse to happen - or sometimes carried it out themselves.
Francis is facing a massive case of abuse in Chile that has led the country’s 34 bishops to offer their resignations.
And Francis will travel soon to Ireland, a country still reeling from abuse scandals. - The Washington Post