Archivist Alexander Singerman, from Shawlands in Glasgow, was last seen on Wednesday 10 May.
He had left his workplace in Auchincruive and had been expected to attend a meeting at the Gaiety Theatre in Ayr, but never arrived.
At about 11.15, a body was found on the nearby Auchincruive Estate. The death is not being treated as suspicious.
Formal identification has not yet taken place, but his family has been advised.
A post-mortem examination is to be carried out to establish the exact cause of death, and a full report will be sent to the procurator fiscal.
Police searching for a 32-year-old man who went missing seven days ago in South Ayrshire have found a body.