We report the first measurement of the tau lepton polarization P-tau(D*) in the decay (B) over bar -> D* tau(-) (v) over bar (tau) as well as a newmeasurement of the ratio of the branching fractions R(D*) = B((B) over bar -> D* tau(-) (v) over bar (tau)) / B((B) over bar -> D* l(-) (v) over bar (l)), where l(-) denotes an electron or a muon, and the tau is reconstructed in the modes tau(-) -> pi(-) v(tau) and tau(-) -> rho(-) v(tau). We use the full data sample of 772 x 10(6) B (B) over bar pairs recorded with the Belle detector at the (KEKB) over bar electron-positron collider. Our results, P-tau(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.