Durst is being held in Louisiana because authorities say they found a gun and nearly five ounces of marijuana in his hotel room when he was arrested in the murder of his friend, Susan Berman, in Los Angeles 15 years ago.
MORE: Read charging document from Superior Court of California
He appeared frail and dazed in a court appearance Tuesday morning.
Durst's attorneys didn't want him sent to the mental health facility, saying they needed him to stay in New Orleans to prepare for future court hearings. But the sheriff appealed, citing health concerns.
Elayn Hunt houses inmates of varying security levels. All seriously or chronically ill inmates in the state go there.
Durst's next court appearance is scheduled for Monday morning.
Durst, who denies having anything to do with Berman's murder, has been charged with capital murder. Prosecutors may pursue the death penalty.
In a documentary about Durst's troubled life, he mumbled about how he "killed them all," providing a dramatic kick to the end of the series. But a law enforcement official said his arrest on the murder charge was based on words he wrote.
Analysis linking a letter Durst wrote to his friend Susan Berman a year before her killing with one he said "only the killer could have written" to point police to her body was the key new evidence in the long-dormant investigation into the 2000 killing, the official not authorized to speak publicly told The Associated Press on the condition of anonymity because the investigation was ongoing.
Durst, 71, was charged Monday in Los Angeles with first-degree murder in the shooting of Berman, the daughter of a prominent Las Vegas mobster. He could face the death penalty under special circumstances that allege he ambushed her and murdered a witness to a crime.
He waived extradition in New Orleans, but authorities there charged him late Monday with being a felon in possession of a gun because he had a revolver when he was arrested there Saturday. Assistant District Mark Burton said they found the pot and gun in his hotel room.
It was not clear how soon he would be returned to California.
Attorney Dick DeGuerin said he wants a hearing in Louisiana as soon as possible to contest the arrest.
"The warrant we believe is based on a television show and not on actual fact," he said. "We want a hearing as quickly as possible so Mr. Durst can go to California and face trial as quickly as possible."
The judge in New Orleans, Magistrate Harry Cantrell, scheduled another hearing for next Monday.
Presented with the two letters in the finale of "The Jinx: The Life and Deaths of Robert Durst" that aired Sunday on HBO, the eccentric heir blinked, burped oddly, pulled his ear and briefly put his head in his hands before denying he was the killer.
Then he stepped away from the tense interview and went to the bathroom, still wearing the live microphone that recorded what he said next.
"There it is. You're caught!" Durst whispered before the sound of running water is heard. "What the hell did I do? Killed them all, of course."
That moment didn't just make for a captivating ending to the documentary on the eccentric life of an heir to a New York real estate fortune, it could also provide additional evidence for prosecutors.
The official who spoke on the condition of anonymity said the bathroom recording was not presented to prosecutors before charges were filed because detectives were still trying to determine if the recording was tampered with in any way.
But legal experts said the audio and other parts of the interview could become key evidence.
"Any statement that the defendant makes that they want to use against him, they can use against him," said Andrea Roth, a law professor at the University of California, Berkeley.
Kerry Lawrence, a defense attorney in Westchester County, New York, said Durst's lawyers will have to try to explain away his comments, perhaps dismissing them as a joke.
"Prosecutors would argue it was a candid moment of self-reflection, and I assume will argue that he knew he was still being recorded, and this was either said in jest or he was being facetious or sarcastic or was being provocative," Lawrence said. "I don't think it's quite the smoking gun."
Durst - still worth millions despite his estrangement from his family, whose New York real estate empire is worth about $4 billion - has maintained his innocence in three killings in as many states.
When Durst approached the filmmakers and agreed to go on camera, against the advice of his lawyers, he had already weathered one murder case, winning an acquittal in a gory Galveston, Texas, dismemberment case by claiming he shot his neighbor in self-defense.
He was still suspected in the killing of Berman, whose father was a Las Vegas mobster associated with Bugsy Siegel and Meyer Lansky, and the disappearance of his wife, Kathleen, who was declared dead long after she vanished in New York in 1982.
Berman, 55, was shot once in the back of the head at her home near Beverly Hills shortly before New York investigators planned to question her about Kathleen Durst's disappearance.
The documentary showed filmmaker Andrew Jarecki confronting Durst with a copy of an anonymous letter that alerted Beverly Hills police to look for a "cadaver" at Berman's address.
Durst offered that whoever sent it was "taking a big risk. You're sending a letter to police that only the killer could have written."
Then, in the final episode, Jarecki revealed another envelope, which Durst acknowledged mailing to Berman, that has similar writing in block letters and also misspelled the address as "Beverley."
The Associated Press contributed to this report
"I wrote this one but I did not write the cadaver one," Durst said. But when shown an enlargement of both copies, Durst couldn't distinguish them.