The second round draw was made at Hampden today and paired Allan Moore's men with Craig Brown's top flight side.
Morton met the Pittodrie men in the same competition in 1996, the last time the two sides clashed.
An incredible encounter in Greenock saw the Dons run out 7-3 winners after extra time, despite having trailed 3-2 with just moments left in the 90 minutes.
The tie is scheduled to be played on either Tuesday 28 or Wednesday 29 August but Aberdeen are due to play a midweek friendly which would clash with the fixture - and say they will now ask the Scottish Football League for the game to be brought forward.
A statement on the Aberdeen FC website said: "Due to our match against Viking Stavanger this league cup match (against Morton) will be played on either Tuesday 21st August or Wednesday 22nd August pending permission being granted by the Scottish Football League."
The full draw is as follows:
ST. MIRREN v AYR UNITED
QUEEN'S PARK v DUNDEE
LIVINGSTON v DUMBARTON
ARBROATH v INVERNESS CALEY THISTLE.
ROSS COUNTY v RAITH ROVERS
MORTON v ABERDEEN
KILMARNOCK v STENHOUSEMUIR
QUEEN OF THE SOUTH v HIBERNIAN
DUNFERMLINE ATHLETIC v MONTROSE
RANGERS v FALKIRK
HAMILTON ACADEMICAL v PARTICK THISTLE