  In view of the recent measurement of nonzero $\theta_{13}$, we carry out a
systematic study of a simple class of neutrino models that has one diagonal
texture or cofactor zero in the mass matrix. There are seven free parameters in
the model and five of them are already measured by neutrino oscillation
experiments; some cases for the normal or inverted hierarchy are excluded and
for the rest we obtain the preferred values for the lightest neutrino mass and
Dirac CP phase. We find that there are strong similarities between one diagonal
texture zero models with one mass hierarchy and one diagonal cofactor zero
models with the opposite mass hierarchy. We also make predictions for
neutrinoless double beta decay for these models. For the one cofactor zero
models, we present a simple realization based on a new U(1) gauge symmetry.
