I don't see why not.  You will most likely have to make up some straps with buckles to attach the MOLLE pack to the ALICE frame, but that shouldn't be too expensive/difficult.  Note I'm just speaking from memory of both systems here; I haven't attempted to do this myself.

Maybe a better question to ask is why one of the MOLLE frames broke.  There are known issues with the MOLLE I and MOLLE II pack frames, and your undamaged frame might not be as strong as you think..  There is a new version out made by the same Mfr which is claimed to have addressed these issues.  Their nomenclature calls it the 1603 type frame.  Some vendors call it a MOLLE Type III frame.  They can be had for about $50 or so.  There are visible differences between new and old in the way the bracing is done, but the reinforcement of the whole thing is not so readily apparent from pictures of the whole frame.  Between the physical changes, and the new-type plastic used, it's a much better frame.

I think I would obtain another frame from the vendor who sold you the cracked unit.  Contact them about doing so, and ask for a pre-paid shipping label so it's on their dime, if possible.  Mention that you'll be detaching the pack before sending the frame, and ask them to carefully examine the frame they're sending you for any damage.
Now, you've gotten what you paid for, and have a fall-back position incase the MOLLE>ALICE reversion doesn't work.  Treat the MOLLE frames with care: don't drop them when the pack is loaded, and be extra careful when it's very cold.  You should be OK.
Eventually the newer type III frames will become more common and cheaper.  You can go that route if you choose at that point.

That's the new frame.  Why they call it Gen IV, I dunno.