  Motivated by recent measurements which strongly support a nonzero reactor
mixing angle $\theta_{13}$, we study a deviation from $S_3$ neutrino discrete
symmetry by explicitly breaking the neutrino mass matrix with a general
retrocirculant matrix. We show that nonzero $\theta_{13}$ and nonzero CP
violation parameter $J_{CP}$ arise due to the difference between $y_2$ and
$y_3$. We demonstrate that it is possible to obtain the experimentally favored
results for neutrino masses and mixing angles from this mass matrix.
Furthermore, we estimate the effective masses $m_{\beta}$ and $m_{\beta \beta}$
and total neutrino mass $\sum |m_i|$ predicted by this mass matrix.
