Emergency services were called to Blackfriars Street at around 3:30 pm yesterday when the 29-year-old took ill.
She was treated on the scene by ambulance staff and was taken to Edinburgh Royal Infirmary where she died.
Police Scotland said they would like to speak to anyone who saw her collapse.
A spokeswoman said: “Police Scotland are appealing for anyone who was in the street around 3.45 and saw a woman collapse in the street to contact us.
”Enquiries are ongoing and officers are keen to speak to anyone who is in the area at this time who should contact police.” Anyone with information is asked to contact police on 101.