A police spokesman said five tortoises, including Hermann's and Indian stars, were stolen from their pens earlier on Sunday, possibly by a visitor who went to the garden which was open to the public.
John Hayward, co-ordinator of the national theft register for exotic animals, said some of the stolen tortoises are on the most endangered species list and can only be bought or sold under licence.
Zeus, a 7st (44kg) Sulcata tortoise, was among the creatures stolen
"Because of their rarity it is illegal to offer such species into trade and it is a most serious offence which can carry upwards of five years in prison," Mr Hayward said.
"It is not the actual monetary value - our main concern is the welfare of these tortoises that have special dietary requirement."
Mr Hayward said the thieves' action had destroyed vital breeding programmes.
Among the animals stolen was Zeus, a 7st (44kg) Sulcata tortoise.
Anyone with any information regarding the stolen tortoises is asked to contact police or Crimestoppers.
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