We report the first measurement of the τ lepton polarization P_{τ}(D^{*}) in the decay B[over ¯]→D^{*}τ^{-}ν[over ¯]_{τ} as well as a new measurement of the ratio of the branching fractions R(D^{*})=B(B[over ¯]→D^{*}τ^{-}ν[over ¯]_{τ})/B(B[over ¯]→D^{*}ℓ^{-}ν[over ¯]_{ℓ}), where ℓ^{-} 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 772×10^{6}  BB[over ¯] pairs recorded with the Belle detector at the KEKB electron-positron collider. Our results, P_{τ}(D^{*})=-0.38±0.51(stat)_{-0.16}^{+0.21}(syst) and R(D^{*})=0.270±0.035(stat)_{-0.025}^{+0.028}(syst), are consistent with the theoretical predictions of the standard model.