The remains have been taken to the police mortuary in Dundee for formal identification and further forensic examinations will take place in the field on Thursday, police said.
Pryde, from Scone in Perthshire, admitted killing Mr Alexander and was jailed for 10 years in April 2010 for what judge Lord Bracadale called a "most disturbing crime".
The long-distance lorry driver was last seen in the High Street near his cottage on November 14, 1999, the day he was killed. He was reported missing by a friend who went to his home four days later and found no sign of him.
The judge heard how Pryde's wife went to police in 2007 after his confession during a drunken row while on holiday in Greece.
Pryde told police that he "just completely lost the plot", grabbed a bar and attacked Mr Alexander. He then dumped the body. "I took him along by the brickworks and buried him. It's never been out of my mind," he added.