Around 40 firefighters tackled the blaze at Powderhall Rigg in the Capital at 9.50am.
On arrival, firefighters discovered a two-storey flat, which is on the third and fourth floor, well alight and smoke billowing from the building. Firefighters in breathing apparatus fought the blaze.
Crews evacuated remaining residents from neighbouring flats.
One male casualty was located but was pronounced dead at the scene.
Five appliances, a height platform, Heavy Rescue Tender and Incident Support Unit were sent to the scene.
A fire investigation team was also mobilised and they have already begun enquiries into the cause of the fire.