Stabilization of the exchange rate comes through increased adoption.

Once a business is using Bitpay (or an equivalent service) to convert instantly convert BTC sales into local currency they will start asking their suppliers to accept BTC directly so the business can benefit from the lower fees associated with keeping the sale in BTC. These suppliers will probably also start using a service like BitPay because their costs are still in local currency.

BTC use then moves up the supply chain one layer at a time, by the time nobody needs national currencies any more the exchange rate will be dominated by the supply and demand for commerce instead of speculation so it will be stable enough to use directly.