Jack was hospitalised on Saturday after a challenge from Motherwell defender Shaun Hutchison in the 2-2 draw at Fir Park.
The 21-year-old was x-rayed there and has been ruled out for up to eight weeks after a further scan revealed the damage.
The news is a sickener for the Dons with the League Cup semi-final at the weekend.
Following the tackle, boss Derek McInnes told the media that Hutchison deserved a red card.
He said: "I thought it was a poor challenge. I know there is a touch on the ball from Hutchison but for me there is only someone going to get hurt in that challenge.
"It is reckless at the very least. In real time I did wince. I thought it was a straight red and having seen it again, I think you would lean more towards a red than a yellow."