The theft, near the Old Chapel, was discovered by borough council staff on Thursday morning (23/8) and has been reported to Dorset Police.
The 36 missing kerb drainage stones are believed to date from the 19th Century.
Cllr Kate Wheller, Brief Holder for Community Facilities at Weymouth & Portland Borough Council, said: “Stealing from the cemetery is like stealing from everyone, it is very sad that someone has done this.
“I would urge anyone who knows anything about this to contact Dorset Police immediately.”
The theft dismayed staff who believe it most likely occurred on Wednesday night (22/8).
The stones are about 30cms (12 inches) square by 12cms (4 inches) high. They are a dark blue in colour like engineers bricks and are shaped to make a water gully. They could possibly be used on a driveway, path or in a garden.
A total of 36 have been taken so a stretch of about 36 foot in length. They have a historical value and cannot be replaced easily. A vehicle would have been needed to move them as they are heavy.
If anyone knows anything about this theft please contact Dorset Police, quoting Crime Number 55180136534 by calling 101 or visiting their website.