Efforts by paramedics to revive a man were unsuccessful.
A statement issued by Greenock Morton said: "It was with deep regret that our match today against Queen of the South FC was abandoned due to the tragic death of one of our loyal supporters during the game."
The club added: "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 Andrew's 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."