Officers were called to the Ferry Road property at about 09:50 on Wednesday.
The baby boy was taken to the Royal Hospital for Sick Children in the city, where he died a short time later.
A Police Scotland spokesman, said: "Police in Edinburgh were called to Ferry Road around 9.50am on Wednesday October 12 following a report of concern for a nine-day-old baby."
He said: "The baby was taken to the sick kids hospital, where he was sadly pronounced dead a short time later.
"Inquiries into the full circumstances surrounding the death are ongoing and a report will be submitted to the procurator fiscal."
The death is being treated as unexplained.
A nine-day-old baby has died in hospital after police were called to an address in Edinburgh.