  We have searched for lepton flavor violating decays of a $\tau$ to a
lighter-mass charged lepton and an $\omega$ vector mesonusing 384.1 $fb^{-}$ of
$e^+e^- $ annihilation data collected with the BABAR detector at the Stanford
Linear Accelerator Center PEP-II storage ring. No signal is found, and the
upper limits on the branching ratios are determined to be ${\cal B}(\tau^{\pm}
\to \mathrm{e}^{\pm} \omega) < 1.1\times10^{-7}$ and ${\cal B}(\tau^{\pm} \to
\mu^{\pm} \omega) < 1.0\times10^{-7}$ at 90% confidence level.
