Mehdi Nemmouche, 29, had filed an appeal against the June ruling that ordered his extradition from France, but his lawyer Apolin Pepiezep said he had decided not to go ahead with the challenge as he now considered the court's decision "satisfactory".
Nemmouche, was arrested on May 30 in the southern French city of Marseille in possession of a Kalashnikov rifle and a handgun similar to the ones used in the attack on May 24.
He was detained on suspicion of murder and attempted murder in connection with a terrorist enterprise after the shooting in Brussels on May 24th that left four people dead.
In June a court in Versailles ordered Nemmouche to be handed over to Belgian authorities in line with the European arrest warrant that had been issued following the murders.
The shooting carried out by a lone gunman, killed three people outright – an Israeli couple and a Frenchwoman, while the fourth victim, a 24-year-old Belgian man, was left clinically dead.
Authorities had released chilling security camera footage of the gunman, wearing a cap and sunglasses, walking into the museum, removing an automatic rifle from a bag and shooting through a door before making an exit.
Belgian media reported that the assailant used a camera to film his attack in the same way as Mohammed Merah, the Frenchman who shot dead several Jews in Toulouse two years ago.
Customs officials detained Nemmouche at Marseille's coach station on board a bus arriving from Amsterdam via Brussels.
According to sources close to the investigation, he was carrying a Kalashnikov automatic rifle and a gun with ammunition in his luggage, as well as a miniature video camera.
"These weapons were of the type used in Brussels," said one source. Another source close to the investigation said that "many elements are consistent with the shooting in Brussels".
The European Jewish Congress immediately drew a parallel between the events in Brussels and the shootings by Merah and called for greater security at Jewish institutions and tougher legislation for dealing with anti-Semitic crime.
Originally from Roubaix in northern France, Nemmouche is believed to have travelled to join Islamist fighters in Syria in 2013, and was known to the French domestic intelligence agency DGSI, the source said.