Mr Durst is waiting to be transferred from Louisiana to Los Angeles to face trial for the 2000 murder of his friend Susan Berman.
The former real estate tycoon was arrested on 14 March - a day before a documentary aired that appeared to show him confess to three killings.
His lawyers say he is innocent and his arrest was illegal.
Mr Durst has spent the much of the past week in a prison's mental health unit.
The 71-year-old millionaire appeared in court wearing padded handcuffs and seated next to his lawyers. Last week, officials deemed him at risk of committing suicide.
Mr Durst is being held in Louisiana on gun and drug charges.
He was arrested at a New Orleans hotel where he had registered under the name Everette Ward.
Mr Durst agreed to be transferred to California to stand trial for Ms Berman's death, but the gun and drug charges have delayed his extradition.
One of the weapons charges alleges that he had a .38-calibre revolver, which he is not allowed to possess after previous felony convictions.
The other charge accuses him of having illegal drugs while in possession of a weapon.
Prosecutors have not said whether they will take those charges to a grand jury.
Police said they found his passport, almost $43,000 (Â£28,700) in cash, the gun, and a latex or rubber mask that could cover his neck and head when they searched his hotel room.
Mr Durst has long been suspected in the disappearance of his first wife, who went missing from their country home in New York state in 1982. He has never been charged.
He was acquitted of the death of his neighbour Morris Black, after claiming the killing was done in self-defence.
He was recently the subject of The Jinx, a multi-part documentary on the American cable network HBO.
The Jinx chronicled his links to his first wife's unresolved disappearance decades ago, the death of Berman and the killing and dismemberment of Black.
In a recording that was made during a break in the filming of the documentary, Mr Durst can be heard saying "I don't know what's in the house," just before saying he "killed them all".
A New Orleans judge has ordered Robert Durst held without bond, saying he is a danger to others and a flight risk.