  We further study a predictive model for the masses and mixing matrix of three
Majorana neutrinos. At zeroth order the model yielded degenerate neutrinos and
a generalized ``tribimaximal" mixing matrix. At first order the mass
  splitting was incorporated and the tribimaximal mixing matrix emerged with
very small corrections but with a zero value for the parameter $s_{13}$. In the
present paper a different, assumed weaker, perturbation is included which gives
a non zero value for $s_{13}$ and further corrections to other quantities.
These corrections are worked out and their consequences discussed under the
simplifying assumption that the conventional CP violation phase vanishes. It is
shown that the existing measurements of the parameter $s_{23}$ provide strong
bounds on $s_{13}$ in this model.
