The Naked Face

A patient of Chicago psychoanalyst Dr. Judd Stevens (Moore) is murdered after a session, stabbed while wearing a raincoat belonging to the doctor. Police detectives McGreavy (Steiger) and Angeli (Gould) investigate. McGreavy bitterly resents the doctor from a previous case in which Stevens' testimony led to a cop killer being institutionalized rather than sent to prison.
The patient's personal problems are in question until Stevens' secretary is also found brutally murdered. Stevens realizes he is the murderer's target, while the detectives treat him as their prime suspect.
A widower, Stevens expresses concern to his physician brother-in-law about the cops' behavior. McGreavy in particular is so aggressive in his interrogations, Angeli reports him to a superior and gets McGreavy thrown off the case.
Posing as doormen, two armed gunmen break into Stevens' apartment. He holds them off until the brother-in-law and two paramedics arrive. Not trusting the cops, Stevens instead made an emergency call to the hospital for help.
Choosing a name out of the Yellow Pages at random, Stevens hires an old, quirky private investigator named Morgens (Art Carney) who lives in a rundown apartment filled with clocks and cats. Morgens saves his life by finding a bomb planted in Stevens' car. After he reports this latest attempt on his life, Angeli promises to lend Stevens a more sympathetic ear than his partner had. McGreavy, however, has secretly cut a deal with their captain to keep an eye on the case from a distance.
Morgens claims to know the killer's true identity, but refuses to reveal it over the phone because nobody can be trusted, not even the police. He tells Stevens to meet him after midnight at Navy Pier and mentions the name "Don Cicini." But when Stevens and Angeli arrive, they find the murdered body of Morgens, holding a cuckoo clock.
After doing some research, Stevens believes that Don Cicini is an Italian code name for the head of an organized crime syndicate. Offering to hide Stevens in a safe house, Angeli instead drives him to a countryside mansion where the phony doormen appear. Angeli draws a gun but points it at Stevens, telling him he trusted the wrong cop.
Stevens is taken to a crime boss, discovering that the mobster's wife is Ann Blake (Anne Archer), a quiet woman who has been seeing Stevens as a patient. Don Cicini feels he simply can't have anyone knowing his private business, despite the fact that Ann told the psychiatrist practically nothing during their sessions.
Stevens is taken to a warehouse to be done away with permanently. Angeli, expecting a payoff, is killed instead. Stevens tries to escape but is re-captured, then beaten savagely by the crime boss until McGreavy and a team of cops arrive just in time, having been called by Ann. Her husband is killed and the rest of his men are arrested.
At a cemetery where Stevens goes to visit his late wife's grave once a week, Ann Blake turns up to apologize for causing Stevens all this trouble at the cost of so many lives. He has quit his private practice, so she asks if that means he can see a former patient socially. It does and they begin to leave the cemetery together, but the doctor is in for one last shock: Ann is shot and killed by a mob sniper, leaving a grieving Stevens wailing over her dead body.

Question: who shot Ann?
Answer: a sniper