I personally think it would be even more popular, since people would be able to convert fiat to Bitcoin and hold them without having to worry about it losing value.
It would be used more as a real payment method than it is now.
The only reason why bitcoin is so popular is because of the price. It changes its price randomly. So if the price of bitcoin is on stable pricenof 500$ for example, would it still be popular compared today ? Thats actually the most special thing about bitcoin ..
What is the most special thing in Bitcoin, its volatility?
If Bitcoin value (not to be confused with its price) is to become rock solid and extremely stable, many merchants would be happy to use Bitcoin a means of payment (just like they use the US dollar right now). So while it could lose some of its popularity among short-term traders and scalpers as well as other speculators, it would certainly gain it among other folks who don't need its volatility so much. Add to this instant payments and you will get an ideal currency (quite close to)