The alarm was raised about the incident in Raeburn Place at 9.54pm on Monday. The fire service confirmed three fire engines and one aerial appliance were sent to the scene.
A spokeswoman for the Scottish Fire and Rescue Service said there had been no reports of injuries.
Footage on social media showed plumes of smoke billowing out from the building.
Diversions were in place for Lothians Buses services 24 and 29 but these have now returned to their normal routes.