  We report on an analysis of $\tau^-$ decaying into $\omega\pi^-\nu_\tau$ with
$\omega\to\pi^+\pi^-\pi^0$ using data containing nearly 320 million tau pairs
collected with the BABAR detector at the PEP-II asymmetric energy $B$-Factory.
We find no evidence for second-class currents and set an upper limit at 0.69%
at a 90% confidence level for the ratio of second- to first-class currents.
