Is it safe to remove peeling asbestos spot and put ceiling tiles over it?

We just bought a house that we love, built in 1962. Because of the age, we were told that there are likely materials with asbestos in the house, but we don't know where.

So there's a spot in the Master bedroom where the popcorn ceiling is peeling, because there was a (fixed) roof leakage. It's an REO, so while they fixed the roof, they didn't bother fixing the ceiling. No damage, just cosmetic. Here's a pic:
Or click this link if it doesn't work: http://img683.imageshack.us/img683/8106/asbestosceiling.jpg

From the research I've read, most people suggest just leaving the popcorn because it's so expensive to get it removed, but in this case, it's damaged already. I think the popcorn is ugly anyway, but I'm not too sure I want to spend thousands to have someone come in and remove it, and delay our moving in in the meantime.

My solution is that I'd like to maybe buy some of those glue-on polystyrene ceiling tiles (fireproof) that can go over popcorn ceilings, and install them ourselves, eliminating the popcorn ceiling threat and saving us some money. We're pretty broke after buying the place itself!

But I've also heard that peeling asbestos ceilings are a threat, because they can release fibers if you mess with them, and ours is peeling in that one spot. Do you think if we wore a mask, sprayed it with water, and just peeled off the small hanging part, it would be safe to install the ceiling tiles over it, or do we have to do the expensive removal?

Any form of glued up anything is not going to work on a popcorn ceiling. You can see how easy it is to have it fall off.
There's lots of companys that can do a simple test to see if it is asbestos. I'd bet it's not though.
Any reason you can not take this down yourself? Then your only cost would be a sprayer, a roll of plastic, a little bit of fabric softner and a drywall knife.
It's a pain and messy but simple to do.
If it's a low texture and you want that stamped metal look you would go over it with plywood then use nail or stape up panels.
The cheapest way would be to go over it with 1/2" sheetrock using 3" long screws after marking were the rafters are.