  We modify the Zee mass matrix by adding a real one parameter perturbation
which is purely diagonal and trace-less. We show that in this way we can
explain both solar and atmospheric neutrino oscillation data. There is a
correlation between the deviation from strict maximality of $|U_{\mu 3}|=
1/\sqrt{2}$, with the emergence of a small but non-zero $U_{e3}$. We calculate
how big a value can $U_{e3}$ get when we restrict ourselves within the allowed
regions of solar and atmospheric neutrino masses and mixing angles. We also
discuss the impact of a $S_2$ permutation symmetry on our mass matrix and show
how a small $U_{e3} \ne 0$ can emerge when this $S_2$ permutation symmetry
between the second and the third generation is broken.
