Both semi-final ties will be played at Hampden
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First Division St Johnstone will have to beat runaway SPL leaders Celtic if they are to reach the Scottish Cup final for the first time.
Hibernian are to play Dunfermline in the other semi-final after the draw was made live on BBC One Scotland.
Celtic's tie with the Saints will be played on Saturday 14 April at 1230 GMT, it has been confirmed.
The second semi-final will be played on 15 April with a 1500 GMT kick-off. Both ties will be played at Hampden Stadium.
Celtic have won the trophy 33 times, while both Hibs and Dunfermline have two triumphs to their names.
Scotland manager Alex McLeish and his assistant Andy Watson were the men responsible for pulling the balls out of the hat.
St Johnstone upset Falkirk and Motherwell en route to the last four, having reached the semis of the CIS Cup earlier this season, with wins over Dundee United and Rangers.
But McLeish told BBC Sport that Celtic would be "firm favourites" for the tie.
Saints manager Owen Coyle said: "Whoever we got, it was going to be hard and they don't come any tougher than Celtic.
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Dunfermline have done ever so well in the cup and they have proved that they can raise their game for the big occasion
Hibernian boss John Collins
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"As a player I always wanted to play against the best and my players will now get that chance.
"It's a big, big ask, but we won't change the way we play. We'll go and make a game of it."
Hibs, who last won the Scottish Cup in 1902, are dreaming of a cup double, having seen off the Saints to reach this month's CIS Cup final, where they will face Kilmarnock.
"On paper, Hibs will be favourites," said McLeish, who had a spell in charge at Easter Road.
"But Dunfermline have had a fantastic cup run, beating Rangers and Hearts, and there is every chance of an upset."
Hibs manager John Collins admitted he was relieved to avoid Celtic but is wary of the SPL's bottom side.
He said: "Dunfermline have done ever so well in the cup and they have proved that they can raise their game for the big occasion."
And Dunfermline midfielder Stephen Simmons, who will be suspended for the tie, was happy with the draw.
"To be honest, we were looking to steer clear of Celtic, so it's a good tie for us," said the former Hearts man.
"But Hibs are a very good side. They beat us 4-0 at East End Park and gave us a real doing."