Can I use a Dremel instead of a purpose made nail drill for acrylics?

I like having acrylic nails, I *hate* spending 2 hours in a salon every couple of weeks and shelling out $35 a trip. So I've starting doing them myself, not quite salon quality but hey for the price ($11/full set) it can't be beat.

I've been looking at the electric nail drills like they use in the salon, I think that would really help give me a better finish (and speed the process up a bit) but I have no desire to spend upwards of $150 on a machine. I was wondering if I could just get a regular cordless Dremel tool and put the nail bits in there. Has anyone ever done this? Will the bits fit? Or am I going to have massive problems?

Failing a Dremel tool, I've been looking on Ebay and found a few decently priced ones but they are all from Hong Kong and I'm wary of buying something like that over Ebay and internationally. Has any one had any good success with going that route?