Edmunds was in City Hospital's intensive care unit Friday night, listed in serious condition.
The Associated Press said Edmunds was being held on a federal warrant charging him with unlawful flight to avoid prosecution for violating probation in Utah on charges unrelated to the Smart case. He has been considered a fugitive since October.
McCabe said the FBI hoped to move Edmunds out of the hospital, but that depended on his condition.
"I don't expect any kind of change or transfer this evening," she said Friday.
An FBI agent in the parking lot at the hospital Friday afternoon referred questions to the FBI's Pittsburgh office.
Edmunds admitted himself to the City Hospital emergency room at 5:15 a.m. Thursday and was admitted to the intensive care unit at 10:30 a.m., McCabe said.
She said Edmunds used an alias when he checked himself in.
Sometime between 1 and 2 p.m. Friday, the intensive care unit staff questioned his identity and suspected that he was Edmunds, McCabe said.
"The FBI was called when there was suspicion that he was not who he said he was," she said.
The FBI arrived within 30 minutes.
In the meantime, federal marshals, West Virginia State Police, Martinsburg Police and hospital security helped safeguard the sixth-floor intensive care unit, McCabe said.
By 3 p.m., the FBI had interviewed the patient and determined he was indeed Edmunds, she said.
Three other patients were moved from the intensive care unit to other parts of the hospital. Authorities locked down the unit, leaving Edmunds as the only patient.
No one from the public or the media was allowed on the floor.
As a result of the lockdown, McCabe said, the hospital stopped accepting new intensive care unit patients, diverting them to other area hospitals.
Late in the afternoon, City Hospital officials were trying to work out a way to open at least part of the intensive care unit.
McCabe did not know how Edmunds arrived at the hospital. West Virginia State Police towed away a green Saturn sedan that Edmunds was thought to have driven.
The car, which had Washington state license plates, was the only vehicle in a section of 12 parking spaces police had cordoned off. It was hauled away on a flatbed truck at about 4 p.m. and the parking lot was reopened.
The car will be searched for evidence of the kidnapping, said Salt Lake City Police Chief Rick Dinse.
The Associated Press contributed to this story.