Referee Don Robertson called time on the match when a fan collapsed on the terraces after suffering a fatal heart attack.
"Due to a tragic incident in the crowd, both clubs have agreed to abandon the match. A date for a replay will be announced in due course," read a message on Morton's Twitter page.
The match was tied at 0-0 when the incident occurred.
Morton released the following statement later this evening:
"It was with deep regret that our match today against Queen of the South F.C. was abandoned due to the tragic death of one of our loyal supporters during the game.
"All at Greenock Morton Football Club would like to take this opportunity to extend our heartfelt sympathy to all family and friends of the gentleman concerned.
"We would further like to register our appreciation for the efforts of the emergency services, St Andrews First Aiders and Stewards who carried out their duties in a professional and caring manner throughout. And finally, our thanks must also go to supporters of both teams who showed patience and great respect as events unfolded."