Mr. Hall also beat Joseph regularly for years before the murder, the judge said on Monday.

Michael Soccio, the chief deputy district attorney, said he hoped the court would get Joseph help. “Joseph is a little boy, and his life has been very, very sad,” Mr. Soccio said after the ruling. But he added: “I also would have been concerned had Joseph been released. I think he’s a very dangerous boy.”

Sitting quietly in court, slight and blond and wearing glasses and a button-down shirt, Joseph showed no visible reaction when the judge found the allegations against him — the murder charge and one count of using a firearm — to be true. He is scheduled to return to court next month, when Judge Leonard will most likely determine where he will be sent.

If he ends up in state custody, Joseph would be the youngest person in the State Division of Juvenile Justice, which houses minors who have committed serious crimes, a state official said.

Matthew J. Hardy, the public defender representing Joseph, said it would be “a complete tragedy” if the boy were sent to state custody. Mr. Hardy said he planned to appeal the ruling.

Mr. Hardy had argued that the environment in the Hall home glorified violence and that, given his young age, left Joseph unable to understand right from wrong. During the trial, Mr. Hardy showed a photograph of Joseph holding a toy gun, giving a Nazi salute and smiling beside a hooded Klansman.

The day before the murder, a reporter attended a neo-Nazi gathering at the Hall home. During the meeting, Joseph sat quietly on the stairs. He said he was having fun, even showing off a new belt bearing a Nazi insignia.

Around 4 a.m. the next day, Joseph pulled his father’s .357 handgun — which he called “the bad gun,” the judge said — down from the shelf in his parents’ bedroom. He walked downstairs, where his father lay sleeping on the couch, and shot Mr. Hall in the head.