Eventually, the difficulty will drop. This will cause the mining to increase. This will cause an increase in the price because the verification problem holding the price down is gone.

There's another factor too -- once you've bought mining hardware, the cost of the hardware is a sunk cost. So long as it can pay for its electricity, you'll keep it running. A typical mining rig today produces Bitcoins worth about 60 times its cost of electricity (1W/GH/s). So the price would have to drop astronomically for people to stop mining.

It could, of course, significantly reduce the incentive for miners to invest in new equipment. But that will just keep the difficulty from rising. It won't significantly slow confirmations.

Also, people might see that the difficulty is going to drop and the price is going to come back up. So they might see this as a great opportunity to get new mining hardware in place in expectation of the easier mining.