We make an analytic and numerical study of leptogenesis in the framework of the (supersymmetric) standard model plus the seesaw mechanism with a U(1) family symmetry and single right-handed neutrino dominance. In presenting our analytic and numerical results Ae make a clear distinction between the theoretically clean asymmetry parameter epsilon (1) and the baryon asymmetry Y-B. In calculating Y-B we propose and use a fit to the solutions to the Boltzmann equations which gives substantially more reliable results than parametrizations previously used in the literature. Our results show that there is a decoupling between the low energy neutrino observables and the leptogenesis predictions. but that nevertheless leptogenesis is capable of resolving ambiguities within classes of models which would otherwise lead to similar neutrino observables. For example we show that models where the dominant right-handed neutrino is the heaviest are preferred to models where it is the lightest and study an explicit example of a unified model of this type.