  We consider the most general neutrino mass matrix which leads to $\theta_{13}
= 0$, and present the formulae needed for obtaining the neutrino masses and
mixing parameters in that case. We apply this formalism to a model based on the
lepton number $\bar L = L_e - L_\mu - L_\tau$ and on the seesaw mechanism. This
model needs only one Higgs doublet and has only two right-handed neutrino
singlets. Soft $\bar L$ breaking is accomplished by the Majorana mass terms of
the right-handed neutrinos; if the $\bar L$-conserving and $\bar L$-breaking
mass terms are of the same order of magnitude, then it is possible to obtain a
consistent $\bar L$ model with a solar mixing angle significantly smaller than
$45^\circ$. We show that the predictions of this model, $m_3 = 0$ and
$\theta_{13} = 0$, are invariant under the renormalization-group running of the
neutrino mass matrix.
