Moira Anderson was 11 when she went missing from her home in Coatbridge, North Lanarkshire, in February 1957 while running an errand for her grandmother.
It is widely believed that she was abducted and murdered, but her body has never been found.
The case remains unsolved, although convicted paedophile Alexander Gartshore, a Coatbridge bus driver who died in 2006, has been connected with the disappearance.
Today, investigators began excavating the plot of Sinclair Upton, said to have been an acquaintance of Gartshore, to see if the schoolgirl's remains were hidden there.
The excavation at Old Monkland Cemetery in Coatbridge is being led by Professor Sue Black and a team from the forensic anthropology department at Dundee University.
Police believe the burial plot has three layers and there may be up to eight people interred there.
Chief Inspector Kenny McLeod, of Strathclyde Police, said: "There may be no necessity to exhume all eight human remains but that does remain a possibility.
"The site does present significant problems and challenges, not only in terms of ground conditions but also record-keeping as you can imagine, going back to 1908, it may not detail every person in the burial plot."
Officers expect to be on the site for the next few days and Prof Black will be in a better position tomorrow to provide a timescale, Mr McLeod said.
Experts are exploring the possibility that Moira's body may have been dumped under a coffin in the grave, which was thought to have been open around the time of her disappearance.
The operation comes after a sheriff gave police the go-ahead last month.
A large section of the cemetery has been cordoned off with police tape, with a large black tent and a series of smaller blue tents erected around the excavation site.
Officers in overalls went in and out of the tents as work started this morning.
Mr McLeod said: "There was a petition before Airdrie Sheriff Court last year from the Moira Anderson Foundation and the Crown Office then instructed Strathclyde Police to facilitate the excavation.
"It comes from new information presented to the court.
"Operations like this are not taken lightly as we remain sensitive to Moira Anderson's family and also the families who have relatives buried elsewhere in the cemetery."
Gartshore has been blamed for Moira's murder by his own daughter, Sandra Brown, in her book Where There Is Evil.