We report the first measurement of the $τ$ lepton polarization $P_τ(D^*)$ in the decay $\bar{B} → D^* τ^- \bar{\nu}_τ$ as well as a new measurement of the ratio of the branching fractions $R(D^{*}) = \mathcal{B}(\bar {B} → D^* τ^- \bar{\nu}_τ) / \mathcal{B}(\bar{B} → D^* \ell^- \bar{\nu}_\ell)$, where $\ell^-$ denotes an electron or a muon, and the $τ$ is reconstructed in the modes $τ^- → π^- ν_τ$ and $τ^- → ρ^- ν_τ$. We use the full data sample of $772 \times 10^6$ $B{\bar B}$ pairs recorded with the Belle detector at the KEKB electron-positron collider. Our results, $P_τ(D^*) = -0.38 ± 0.51 {\rm (stat.)} ^{+0.21} _{-0.16} {\rm (syst.)}$ and $R(D^*) = 0.270 ± 0.035{\rm (stat.)} ^{+0.028}_{-0.025}{\rm (syst.)}$, are consistent with the theoretical predictions of the Standard Model. ; Comment: 7 pages, 3 figures, submitted to Physical Review Letters