It has long been suspected that the tomb, in an underground crypt, may contain not only the body of the mobster but also the remains of 15-year-old Emanuela Orlandi, who is thought to have been kidnapped by his gang in June, 1983.
The saga of Miss Orlandi's disappearance is one of the murkiest intrigues to have embroiled the Vatican in modern times.
Sub-plots involve the attempted assassination of Pope John Paul II, the imprisonment of his Turkish assailant, far-Right political groups and "God's banker", Roberto Calvi, who was found hanging beneath Blackfriars Bridge in London in 1982.
The alleged mastermind behind the kidnapping of Miss Orlandi was Enrico "Renatino" De Pedis, the leader of the Magliana gang, Rome's most ruthless criminal group.
He was shot dead by rival gangsters in a street in central Rome in February, 1990, and his body interred in the crypt in Basilica of Sant'Apollinare, a Vatican-owned church near Piazza Navona.
Only his widow, Carla Di Giovanni, was said to possess the keys to the sarcophagus. In 2005, an anonymous caller to the Italian equivalent of the BBC's Crimewatch programme suggested that investigators hoping to solve the mystery of the teenager's disappearance should open the crypt.
The man who made the call has since been identified as the son of a member of the Magliana gang.
Five years on, the Vatican, through the Vicariate of Rome, has finally given police and magistrates permission to prise open the marble crypt and analyse its contents.
"The Vicariate of Rome expresses no objection to the request of Italian magistrates that the tomb of Signor De Pedis be inspected," the Vicariate said in a statement.
Church officials also consented to the corpse being removed from the crypt, in the hope that it could shed some light on the teenager's disappearance.
Police are interviewing witnesses in the case, which was reopened two years ago, and are unlikely to open the tomb until next month or in September.
It has always been seen as highly unusual that a known mafioso should have been given the honour of being buried in a church in which popes and cardinals are also buried.
The church is next to the music school which Miss Orlandi attended and where she was last seen alive, climbing into a dark green BMW with an unidentified man.
The teenager's family said they were sceptical over claims that her remains might be in the tomb.
"We will await the facts," said her sister, Natalina Orlandi. "But I have never believed that Emanuela could be buried with De Pedis.
"As a Catholic, to think that an underworld boss could be laid to rest in a basilica is upsetting."
Over the years it has been claimed that Miss Orlandi was kidnapped so that she could be used as a bargaining chip for the release from prison of Mehmet Ali Agca, the Turkish gunman who shot and wounded Pope John Paul II in St Peter's Square in 1981.
Her parents received anonymous phone calls in which it was proposed that their daughter would be released in exchange for Agca, who was a member of an ultra-Right Turkish group called the Grey Wolves and allegedly carried out the assassination attempt on the orders of Bulgarian intelligence and the KGB.
Agca served 19 years in jail in Italy before being deported to Turkey, where he was released in January.
Another theory is that the girl's father, a Vatican employee, had stumbled on documents that connected the Vatican bank with the Magliana gang and that she was kidnapped in an attempt to silence him.
It has even been suggested that the kidnapping was carried out on the orders of a Catholic Archbishop, Paul Marcinkus, the disgraced head of the Vatican bank, known as the Istituto per le Opere di Religione.
Marcinkus, an American who died four years ago, was accused of having links not only with Roberto Calvi but with organised crime groups.
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