The Hoops were due to face the league's seventh-place side at 3pm on Saturday.
But Celtic Park is unable to host the fixture due to the affects of Storm Eowyn on the stadium.
Central Scotland was among many areas badly affected by winds of up to 100mph on Friday.
The Met Office issued a red weather alert signalling danger to life.
At 4pm on Friday, fans were warned the match could be called off - with extreme conditions preventing staff from assessing the damage at Parkhead until Saturday morning.
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Now, five hours before kick-off, the game has officially been postponed.
A Celtic statement read: "Unfortunately, due to the stadium damage sustained yesterday we have been advised that we are unable to proceed with today's match.
"While clearly this is disappointing, the safety of our supporters will always be our priority.
"The rearranged fixture will take place on February 5, 2025 with a 7.45pm kick-off."
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It comes after footage showed Storm Eowyn ripping off part of the roof at Rangers' Ibrox stadium.
The Scottish League One clash between Arbroath vs Kelty Hearts was called off on Friday, while Ross County vs Hibs is set to go ahead after a pitch inspection at 9.45am on Saturday.
Meanwhile, the weather conditions have reportedly affected Celtic's attempt to re-sign Jota.
The Portuguese forward, who left Celtic in 2023 for £23million, is returning to Parkhead in a deal which sees Kyogo Furuhashi go in the opposite direction.
However, according to Sky Sports, his flight to Glasgow on Friday was cancelled.
Kyogo did make it to France, where he is finalising a £10m move to the Ligue 1 side.
Meanwhile, it is understood Jota will complete a £8m return to Celtic over the weekend when flights resume.
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