The coroner's office said David Vopat, 51, of Chino, died in the crash on Tuesday.
Vopat was flying a Northrop N9M built in 1944 and restored by the Planes of Fame Air Museum in nearby Chino.
Vopat was an experienced pilot and multiple YouTube videos produced by the museum show him handling the aircraft.
The aircraft, also known as the Flying Wing, crashed around noon on the grounds of the California Rehabilitation Center on Fifth Street in Norco.
The plane burst into flames after the crash and very little was left intact after the fire was extinguished.
The museum's website says the plane was considered a forerunner of the modern B-2 stealth bomber.
Northrop built four prototypes, but the one that crashed had been the only one still remaining, the museum said. The museum completed restoration of the aircraft in the 1990s and completed flight testing in 1996.
Its specs say it was just over 17 feet long and could hit a top speed of 220 mph.
A friend of the pilot said he was also the aircraft's mechanic and he was preparing for a May 4 airshow.
The cause of the crash remains under investigation.