It happened at the facility in the 4200 block of Monument Road just after 12:00 a.m. on Thursday.
Police say one male patient attacked another male patient.
The 23-year-old victim was taken to Hahnemann Hospital where he was pronounced dead.
Homicide detectives are questioning the man who, they say, carried out the attack.
So far, police have not classified the case as a homicide and no charges have been filed.
The 147 bed Belmont Behavioral Hospital is a private psychiatric facility that treats a number of behavioral disorders, including addictions.
Officials at the hospital say they are fully cooperating with the investigation.