The incident involved a car and a pedestrian and took place on Easter Road in the early hours of Saturday morning, June 19.
The pedestrian, a 21-year-old man, was taken to hospital, and died on June 29. His family has asked for privacy at this time and the victim has not been named at their request.
Police Scotland said the delay between the man’s death and the details being made public was due to issues informing relatives.
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A 23-year-old man has been arrested in connection with the incident.
Sergeant Paul Ewing, from the Edinburgh Road Policing team, said: “At this time, our thoughts are very much with the family and friends of the deceased.
“We are continuing our enquiries into the incident and would ask anyone who was in the area at the time and may have information to come forward.
"We’re also particularly keen to speak with two females who were seen near to an ATM at the bottom of Easter Road, around 2.30am.”
Anyone who can help should call 101, quoting incident 0590 of Sunday, 19 June, 2022.