  We constructed a model of neutrino masses using Froggatt-Nielsen mechanism
with $U(1) \times Z_3 \times Z_2$ flavor symmetry. The model predicts that
$(2/3)m_2/m_3 \sim \sqrt{2}\sin\theta_{13}$ at lepton number violating scale
$M_1$. It is shown that the small values for $m_2/m_3$ and $\sin\theta_{13}$
are consequences of breaking discrete symmetries.
