The 34-year-old man was rushed to hospital after becoming unwell during the weekend bash, but doctors were unable to save him.
Three ambulances were called to the flat at Porteous’ Pend just after 9am on Sunday morning after the man took ill on the landing outside the second-floor flat.
Paramedics treated him at the scene before speeding to Edinburgh Royal Infirmary, where he later died.
It’s understood that the man – who has not yet been named – had been a guest at a flat party on Saturday night which carried on into the early hours of Sunday.
The quiet close of Porteous’ Pend, in the shadow of Edinburgh Castle, is tucked away from the busy Grassmarket near the Apex International Hotel, and contains several blocks of tenement flats.
A neighbour, who did not want to be named, said she had heard the party and people “going up and down the stairs” but there had been “no disturbance”.
She added: “It was only because I got up early that I realised. The police arrived and sealed off the flat.
“They have had the odd party but there’s been no trouble. This is very unusual for this stair – it’s really quiet. We have got good neighbours.”
The couple who were hosting the party are believed to be young professionals in their 30s who have lived there for several years.
Yesterday, the door of the rented property – which is one of six flats in the stairwell – had been secured with a steel board. Shutters were closed on one window, while a yellow flag with a “smiley face” on it could be seen hanging from the outside ledge.
One neighbour said ambulance workers had given the man oxygen before taking him to hospital. “It’s a shame, I’m really surprised,” she added.
Police said they were investigating the “unexplained” death but it is understood that there are no suspicious circumstances.
A spokeswoman said officers had been called “to assist the ambulance with a male having a cardiac arrest”.
Police Scotland added: “Police are investigating after a 34-year-old man was taken ill at an address in the Grassmarket at around 9.10am on Sunday.
“He was conveyed to ERI but was later pronounced dead. Police are treating his death as unexplained and inquiries are at an early stage.”