Rangers will host Old Firm rivals Celtic and holders Aberdeen face a potential trip to Dunfermline in the Scottish Cup quarter-finals.
The draw, conducted following Falkirk's extra-time win at Stenhousemuir, means the Pittodrie club take on Neil Lennon's Scottish Championship side if they can see off Motherwell.
Elsewhere, St Mirren will host Partick Thistle while Dundee United or Spartans will visit Falkirk.
The ties will be played on the weekend of Saturday 7 March.
Dundee United's meeting on Tayside with Spartans was postponed due to a waterlogged pitch, with the tie taking place on Tuesday 17 February (19:45 GMT).
Motherwell's trip to take on Aberdeen, who beat Celtic in last year's final, is the following day (19:45).

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