  We make an analytic and numerical study of leptogenesis in the framework of
the (Supersymmetric) Standard Model plus the see-saw mechanism with a U(1)
family symmetry and single right-handed neutrino dominance. In presenting our
analytic and numerical results we 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
parametrisations 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.
