The Paisley side are 10th in the Scottish Premiership, above Kilmarnock on goal difference and three points below the Dons.
After the league trip to Celtic on 11 April, McLeish’s side will face the same opponents in the Scottish Cup semi-final at Hampden on 19 April.
“I’m delighted to be given the opportunity to lead the group for the rest of the season,” McLeish told club media.
“We’ve got a lot of big games to look forward to, including the cup semi-final, and I can’t wait to take part in that opportunity as well.
“We spoke to a number of different people and Stuart ticked all the boxes with his level of experience. He’s been down in the English Premier League, has managed himself, is very good on the grass and on the coaching pitch, and is ultimately a really good person and good about the place. Everyone I spoke to spoke really highly of him.”