ED Harris gives another terrific performance as a priest suffering from a crisis of faith in “The Third Miracle,” a flawed drama offering a rare look at the Catholic Church’s canonization process.

Harris plays Father Frank Shore, a “postulator,” or clerical detective, whose job is to investigate miracles attributed to candidates for sainthood.

As the movie opens, Frank is living on the streets of Chicago as a lay person, his spirit in disarray after he exposed one candidate as a suicide — an episode that left him with the nickname “The Miracle Killer.”

Frank’s superiors track him down and hand him a new job. In a poor parish, a deceased immigrant woman is being credited with a bleeding statue of the Virgin Mary, as well as the miraculous cure of a girl suffering from a brain tumor.

Father Frank very reluctantly tracks down the girl, who has grown up into a prostitute (Barbara Sukowa), as well as the dead woman’s agnostic daughter (Anne Heche), with whom he is drawn into a none-too-convincing romance.

The skeptical cleric comes to believe in the miracles, but needs proof of a third one. He learns that during World War II, the woman lived in an Austrian village where the Allied bombing abruptly stopped — after she prayed to a statue of the Virgin Mary.

The movie climaxes with a trial that pits Father Frank against a cardinal (a hammy Armin Mueller-Stahl) who serves as a devil’s advocate against the canonization — and it’s here where the movie self-destructs with a mind-boggling coincidence.

Agnieszka Holland (“A Little Princess”), who previously directed Harris in “To Kill a Priest” (in which he played a Polish secret police officer) lays out the story at a rather leisurely pace. It’s not terribly clear when it’s taking place; Heche’s fake-fur-trimmed coats suggest the 1970s.

“The Third Miracle” is worth seeing mainly for Harris and its unusual subject matter.