The Argentine, who has netted six goals in 11 league games this season, was preparing for this weekend's Bundesliga clash at Hamburg when he sustained the problem on Wednesday.
MRI scans on Thursday revealed the extent of the injury, meaning Di Santo will be missing for Sunday's Nordderby.
"This is obviously a setback for us," said coach Viktor Skripnik. "Franco was in very good shape, we would have liked to have had him here.
"But we have good alternatives who have our full confidence.
"I am sure that we will compensate for the loss together."
Ludovic Obraniak is also expected to miss the derby fixture with a thigh injury, while Sebastian Prodl will be assessed during training on Friday following his return from international duty with Austria.