It was found as part of a cataloguing programme, which has been examining about 15,000 animal bones excavated from the cavern that had been housed in the museum's store.
The museum's researchers found the butchered bone in June, and, working with the University of Oxford's School of Archaeology and Radiocarbon Accelerator Unit, identified it as a fragment of human arm bone.
It was then radiocarbon dated to 8,185 years BP [Before Present, an archaeological term meaning before 1950].
Tom Higham, from the radiocarbon unit, said: "The bone was particularly well preserved and the result is seen as very reliable."
Dr Rick Schulting, of the University of Oxford's School of Archaeology, said: "Finds like this highlight the complexity of mortuary practices in the Mesolithic (Middle Stone Age), many thousands of years before the appearance of farming, which is more usually associated with complex funerary behaviour."
The museum said only one other site in Britain had yielded similar human remains with cut marks of this age - Gough's Cave at Cheddar Gorge.
"Some archaeologists have interpreted these marks as evidence of cannibalism, but ritual burial practice or dismemberment for transportation has not been ruled out," a museum spokesman said.
Archaeological digs there have unearthed a 37,000-year-old human jawbone and stone tools that were more than 40,000 years old.
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