The death is currently being treated as unexplained though it is not being treated as suspicious.
Paramedics alerted the police at around 5:35am after discovering the man lying on Leamington Terrace near to its junction with Bruntsfield Place.
Detective Inspector Alan O’Brien thanked members of the public came forward and helped with inquiries.
He added: “Following a public appeal we successfully traced a number of witnesses. At this stage of our investigation there are no suspicious circumstances although the exact cause of death is yet to be determined.
“We have been in contact with the next of kin and will continue to offer them support at this extremely difficult time.
“Anyone who has any information relevant to this investigation is asked to contact police on 101 or the charity Crimestoppers on 0800 555 111.”
The next of kin have been informed but it is understood that they have asked police not to release the name of the man.