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.

Up to 25 people are being held, a top official told the BBC. Police are also reportedly hunting several students.
Espousing separatism is illegal in Sri Lanka.  Suspects can be held without charge for 18 months.
The detentions come at a time of rising political tensions in the north.
Last week security forces and students clashed at Jaffna University in the worst political disturbances since the civil war ended in 2009. Students had tried to mark "Martyrs' Day", which commemorates dead Tamil Tiger fighters.
Officials gave differing accounts of the number of arrests made by anti-terrorist police in the latest raids.
Police spokesman Prishantha Jayakody said that 10 men had been detained on Wednesday in the Jaffna area, all of them accused of "terrorist activities" with the Tamil Tigers or LTTE before 2009.
Mr Jayakody said that none those being held had completed a government rehabilitation course for LTTE suspects and none had surrendered to the police or army at the end of the war.
"They were arrested at different places on the Jaffna peninsula and are now in Vavuniya," he said.  "All the families have been informed."
He said that police were guided to the suspects by intelligence reports.
But the head of the police's Terrorist Investigation Division, Chandra Wakista, told the BBC that as many as 20 to 25 had been detained.
"No arrests have been made without procedure, all of them were made officially," he said.
The police say families have been given official documents relating to the arrests.
But the BBC's Charles Haviland in Colombo says that some relatives have apparently been kept in the dark - one family told a newspaper they only got official information on their son's arrest a long time after he disappeared, and only after they informed the official human rights body.
Rights workers are also mindful of the fact that a year ago two young campaigners disappeared in Jaffna, where tensions have continued since the end of the war in May 2009.
Separately, sources in Jaffna University say the police have given them a list of 10 students they want to arrest.
Three students already being held are said to be suspects in a petrol-bombing and in putting up pro-LTTE posters.

Joseph Travers will be charged with gross negligence manslaughter in relation to the death of Ryan Harvey and misconduct in public office.
Mr Harvey, 23, was found hanged in his cell at the Buckinghamshire jail on 8 May 2015.
Mr Travers is thought to be the first prison officer to be charged with manslaughter after a prisoner death.
The Crown Prosecution Service said: "Following careful consideration of all the evidence provided to us by the police it has been decided that Joseph Travers should be charged."
Mr Harvey was one of five prisoners found hanging in their cells in HMP Woodhill in 2015.
The prison had the worst record for suspected suicides in all of England and Wales that year.
Mr Travers will appear before Westminster Magistrates' Court on 20 April.
HMP Woodhill, a Category A prison, opened in 1992 and can hold 819 inmates.
A nine-day-old baby has died in hospital after police were called to an address in Edinburgh.
[NEXT_CONCEPT]
Police in northern Sri Lanka have made a series of arrests which they say are linked to combating terrorism, three years after the separatist Tamil Tigers were defeated.
[NEXT_CONCEPT]
A prison officer is to be charged with manslaughter following the death of an inmate at HMP Woodhill, Milton Keynes.