Mr Scott, 90, was reported missing on Tuesday, December 11th, from the Chesser area of the city.
He was later found on Friday, January 4th, in the Water of Leith near to Victoria Bridge at Leith Docks.
Police said the death is being treated as non-suspicious and a report will be submitted to the Procurator Fiscal.
Chief Inspector Jordana Emerson, Area Commander for South West Edinburgh, said: “William’s family has shown considerable bravery and composure throughout this harrowing time and our thoughts are with them as they come to terms with their loss.
“I would like to thank those members of the public and media who assisted in our search, their support was very much appreciated.
“We are supporting William’s family and will continue to work with them and offer assistance as required.”