The skeleton was found in a shallow grave as work was carried out on the museum’s drains.
The remains have been removed by archaeologists. They will be examined by experts to try to work out how long the skeleton had been there for.
The museum was built on the site of the medieval St Mary’s Abbey. A 4m (13ft) deep well was also discovered.
Its curator of archaeology, Andrew Morrison, said: “At this stage it is very hard to work out much about him or her, but they were buried east to west which suggests a Christian burial.