So they find ways to encourage one another. They share books like Father James Martin’s groundbreaking “Building a Bridge,” on the relationship between the Catholic and L.G.B.T. communities. Some have signed petitions against church-sponsored conversion therapy programs, or have met on private retreats, after figuring out how to conceal them on their church calendars. Occasionally, a priest may even take off his collar and offer to unofficially bless a gay couple’s marriage.

Some may call this rebellion. But “it is not a cabal,” one priest said. “It is a support group.”

Just over a year ago, after meeting with a group of gay priests, Father Greiten decided it was time to end his silence. At Sunday Mass, during Advent, he told his suburban parish he was gay, and celibate. They leapt to their feet in applause.

His story went viral. A 90-year-old priest called him to say he had lived his entire life in the closet and longed for the future to be different. A woman wrote from Mississippi, asking him to move south to be her priest.

To some church leaders, that outpouring of support may have been even more threatening than his sexuality. Father Greiten had committed the cardinal sin: He opened the door to debate. His archbishop, Jerome E. Listecki of Milwaukee, issued a statement saying that he wished Father Greiten had not gone public. Letters poured in calling him “satanic,” “gay filth” and a “monster” who sodomized children.

“We have to get it right when it comes to sexuality.”

The idea that gay priests are responsible for child sexual abuse remains a persistent belief, especially in many conservative Catholic circles. For years, church leaders have been deeply confused about the relationship between gay men and sexual abuse. With every new abuse revelation, the tangled threads of the church’s sexual culture become even more impossible to sort out.

Study after study shows that homosexuality is not a predictor of child molestation. This is also true for priests, according to a famous study by John Jay College of Criminal Justice in the wake of revelations in 2002 about child sex abuse in the church. The John Jay research, which church leaders commissioned, found that same-sex experience did not make priests more likely to abuse minors, and that four out of five people who said they were victims were male. Researchers found no single cause for this abuse, but identified that abusive priests’ extensive access to boys had been critical to their choice of victims.

The notion that a certain sexual identity leads to abusive behavior has demoralized gay priests for decades. Days after one man retired, he still could not shake what his archbishop in the 1970s told all the new priests headed to their first parish assignments. “He said, ‘I don’t ever want you to call me to report about your pastor, unless he is a homo or an alchie,’” he said, referring to an alcoholic. “He didn’t even know what he meant when he said homo, because we were all homos. He meant a predator, like serial predator.”