They could turn me down, and I'm trying very hard to believe that they will, because it's less disappointing if you don't have high expectations. But...
They haven't advertised any positions. They said they were going to anyway, since they are growing, and said it was very convenient that I turned up at the time I did.
They also said that if I do get the position, I will start pretty much immediately, so it looks like I'd be bypassing the usual hiring process since they already know me.