Holders Celtic have been drawn away to Peterhead in the fourth round of the Scottish Cup as Auchinleck Talbot were handed their biggest-ever tie.
Scottish Premier League leaders Rangers are away to Second Division title contenders Arbroath.
Inverness Caledonian Thistle host Dunfermline Athletic in the only tie between two top-flight sides.
And Junior Cup winners Auchinleck have been handed a tie away to SPL outfit Hearts.
Talbot, who beat East of Scotland side Vale of Leithen 3-1 in the previous round, have taken about 6,000 fans to recent Junior Cup finals.
And club treasurer David Loy told BBC Scotland: "I think there will be more going to this than a Scottish Junior Cup final.
"It is the game of the century for us."
Commercial manager Colin Chisholm admitted there were mixed feelings in the Auchinleck camp about the draw.
"It's not the ideal fixture the manager wanted," he said. "His preference was for a lower-league club at home first and hopefully get through that and get a major club.
"But it's a great occasion as far as the club's concerned and also the supporters.
"I know we have been in 10 Junior Cup finals, of which we were successful in nine, but nonetheless to go into a game of this magnitude against such a club and in this competition is a tremendous thing.
"From a financial point of view, it is a very lucrative fixture.
"Hopefully, we can look at doing more ground improvements, which will benefit everyone in the area."
Chisholm does not discount a surprise victory for the Ayrshire side against a club who are having their own financial problems.
"If the players play to their ability, strange things happen in football," he said.
"Several of our players have had senior experience and will be looking forward to going back in there against a senior club."
The other surviving Junior side, Culter, who were reinstated into the competition after Spartans played an ineligible player, will host Queen of the South should they win their replay away to another First Division outfit, Partick Thistle.
Celtic first-team coach Garry Parker was not taking anything for granted despite the Glasgow side being handed a trip to face a team sitting second bottom of Division Three.
"We'll have them watched and do our homework," he said of the match-up with the side managed by former Celtic midfielder Jim McInally.
"He obviously knows a lot about Celtic."
Parker is hopeful that the management team led by Neil Lennon can match last season's feat, when they defeated Motherwell in the final.
"First year we have been here and go and win the trophy and hopefully we can go all the way again," he said.
Kilmarnock have been handed a tough draw away to First Division side Dundee, while three other SPL sides face ties away to lower-league opposition.
Dundee United take on Second Division strugglers Airdrie United, Aberdeen await the winners of replay between Third Division outfit Stranraer and Division Two's Forfar Athletic, while Hibernian face Division Two leaders Cowdenbeath.
Cowden manager Colin Cameron welcomed the draw.
"The guys played three Pemier League teams in pre-season and enjoyed it," he said.
"It is a stage for them to try to prove themselves. They are capable and I've got belief in them.
"And, with the money the tie will generate, it will keep the club afloat."
The remaining three SPL clubs have home ties.
Last season's beaten finalists, Motherwell, await the winner of the all-Division Three replay between Elgin City or Queen's Park.
St Mirren take on First Division side Hamilton Academical, while Second Division outfit Brechin City face a trip to Perth to face St Johnstone.
Scottish Cup fourth-round draw
Airdrie United v Dundee United
Arbroath v Rangers
Cowdenbeath v Hibernian
Culter or Partick Thistle v Queen of the South
Dundee v Kilmarnock
Falkirk v East Fife
Hearts v Auchinleck Talbot
Inverness Caledonian Thistle v Dunfermline Athletic
Livingston v Ayr United or Montrose
Motherwell v Elgin City or Queen's Park
Peterhead v Celtic
Raith Rovers v Morton
Ross County v Stenhousemuir
St Mirren v Hamilton Academical
St Johnstone v Brechin City
Stranraer or Forfar Athletic v Aberdeen
Ties to be played 7/8 January