LNER, who run Waverley, said new customers would be turned away.
Huge queues were reportedly in evidence as people heading south tried to catch the last train.
A spokesperson said: “Please DO NOT TRAVEL unless absolutely necessary. Tickets will be valid all day tomorrow.”
The Edinburgh Evening News reports that police have been deployed at all of the station’s entrances, preventing people from getting in.
Meanwhile trains on the main Edinburgh Glasgow rail route via Falkirk High were subject to disruption or cancellation today.