I started a relationship about a year before I left for grad school. For the first year of school we were on opposite coasts, and now we're a couple hours apart. Obviously it's not easy but if you know you want to be committed to that person then you make it work. I've known people who are on different continents and still make it work. Assuming the girl knows that you will be gone in three months, I don't see why you shouldn't see where the relationship leads. Maybe it will work out and maybe it won't, but you might regret it if you don't at least try.