Losing a capital is not the end of the game, though if it happens early enough it's pretty much a death sentence. Most human opponents in multiplayer will surrender if you manage to do this, but the AI will often fight on in its hopeless situation. However, if you've come this far, it shouldn't be too hard to "clean up".
Without a capital, all the planets in the empire will slowly lose loyalty until reaching 25% (culture can still bump this up to 35%). This will kill all their income except their refineries and trade ports. To get their loyalty back, they will need to establish a new capital. This can be done on any type of planet, but I wouldn't recommend doing it on asteroids. It's highly expensive, as well.
As I said, it's not the end of the game, but if you manage this early enough you shouldn't have any problem finishing off the AI shortly afterwards.