Ally McCoist's men face a trip to Airdrie to play the amateurs, whose Hampden home is being used for the Commonwealth Games.
The date is yet to be fixed as Rangers organised a friendly in Derby on the August 2 first-round date because of the Glasgow 2014 Games. They start their season with a Petrofac Training Cup tie against Hibs on August 5.
Championship winners Dundee host Peterhead while fellow promoted side Hamilton will be at home to Arbroath.
Relegated Hearts and Hibs enter the competition in the second round.
Queen of the South v Elgin City
Brechin City v Dumbarton
Dunfermline Athletic v Annan Athletic
Queen's Park v Rangers
Morton v Berwick Rangers
Montrose v Falkirk
Raith Rovers v Forfar Athletic
Albion Rovers v Livingston
Hamilton Academical v Arbroath
East Stirlingshire v Ayr United
Dundee v Peterhead
Clyde v Cowdenbeath
Stranraer v East Fife
Airdrieonians v Stenhousemuir
Alloa Athletic v Stirling Albion
Ties are due to be played on Saturday 2 August with the exception of Queen's Park v Rangers, which will be rescheduled due to Glasgow 2014.