The 62-year-old was reported missing from his St Cyrus home after last speaking to his family in Dundee on 14 July.
Mr Leslie's body was found on Saturday morning.
Police Scotland said there were no suspicious circumstances surrounding his death and that Mr Leslie's family would not be issuing a statement.
Insp Megan Heathershaw said: "Our thoughts are with Mr Leslie's family and friends at this very sad time.
"Following our appeals for information, numerous members of the public got in touch to assist with information.
"I would like to take this opportunity to thank these people for their help."
Police have confirmed that a body discovered in a village near Montrose is missing man Graeme Leslie.