Mold prevention over concrete slab
This starts to get into a gray area. For your specific situation, you could use a product called RedGard. It comes in a liquid form and would get rolled over the slab. Once it's dry, it will be a moisture barrier. Here's where it gets tricky. You CANNOT put any type of material over this that can't breathe. If you used some kind of moisture-proof or vapor barrier on top of the RedGard, you'd create a moisture sandwich and have the same problem all over again. As long as you use padding that breathes, you can carpet again and your problem should be solved. However, don't plan on putting some kind of vinyl product or a floating floor w/vapor barrier on that floor ever.