A Catholic high school in Indiana said Sunday that it is firing an openly gay teacher to avoid “forfeiting our Catholic identity” — the second school in the Archdiocese of Indianapolis to publicly reckon with the issue of homosexuality in a week.

In an open letter, officials with Cathedral High School in Indianapolis said the teacher, who is in a same-sex marriage, was “living in contradiction to Catholic teaching.”

To remain within the archdiocese, the letter said, “Cathedral must follow the direct guidance given to us by Archbishop Thompson and separate from the teacher.”

If the school were to continue employing the teacher, the letter said, Cathedral would no longer be considered a Catholic school. The letter said Cathedral would lose its nonprofit status, its priests would not be able to serve on the school’s board of directors and it could not celebrate important Catholic rituals.

The letter, which was signed by Matt Cohoat, chairman of the school’s board of directors, and president Rob Bridges, described the firing as an “agonizing” decision.

The move was in contrast to another school, Brebeuf Jesuit Preparatory School, that defied an order from Archbishop Charles Thompson last week to fire a long-time teacher in a same-sex marriage.

In that case, school officials said Thompson’s “direct insertion into an employment matter of a school governed by a religious order is unprecedented.”

“After long and prayerful consideration, we determined that following the Archdiocese’s directive would not only violate our informed conscience on this particular matter, but also set a concerning precedent for future interference in the school’s operations,” school officials said.

The archdiocese cut ties with Brebeuf after its announcement, The Indianapolis Star reported.

In its letter, Cathedral, which is run by a different order, the Brothers of the Holy Cross, said that while it “respected” Brebeuf’s decision, it could not continue to function in the same way as that Jesuit-sponsored school were it to be banished by the archdiocese.