We present a study of the CP asymmetry in the B⁰ → D*+D*- decay, using data produced by the electron-positron asymmetric-energy accelerator KEKB (Tsukuba, Japan) running at the Υ(4S) resonance. We use the full data sample (as of summer 2008) of 657 × 106BB events, which corresponds to an integrated luminosity of 605 fb-1 recorded with the Belle detector. From the distributions of the proper-time intervals between a neutral B meson decay to the D*+D*- final state and another flavor-tagged B⁰ or B⁰ decay, we measure the CP-violation parameters A and S. A is a measure of the direct CP violation in the decay amplitudes while a non-zero S value indicates CP violation in the interference between the decay and the mixing amplitudes. At the quark level the B⁰ → D*+D*- decay is a b → ccd transition, where the tree amplitude is Cabbibo-suppressed. The contribution of penguin amplitudes in this decay is estimated to be at the percent level . When ignoring penguin corrections, the SM prediction for the CP parameters are AD*+D*- = 0 and SD*+D*- = -ηD*+D*- sin 2β, where ηD*+D*- is the CP eigenvalue of D*+D*- and β = arg[-VcdV*cb]/[VtdV*tb] is an observable phase related to the coefficients Vij of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix. Any large measured deviation with respect to prediction can be a sign of New Physics. As the D* particle is a vector-meson three different partial waves contribute to the final states: the S and D correspond to the CP-even state, while the P wave corresponds to a CP-odd state. To avoid too large a dilution of the CP asymmetry, the two CP components should be separated on a statistical basis. We measure the CP-odd fraction by performing a time-integrated angular analysis in the transversity basis and obtain: R⊥ = 0.125 ± 0.043(stat) ± 0.023(syst). which is consistent with the prediction from theory as well as the previous measurements. We have taken into account the CP-odd dilution and the effect of incorrect flavor assignment to model the decay rates. The proper-time difference of the two B mesons is obtained using the reconstructed D-meson trajectories and the beam spot profile. A "null" measurement is performed by fitting the CP asymmetry of the control sample, B⁰ → D(*)+Ds(*)- which gives A = -0.02 ± 0.03(stat) and S = -0.07 ± 0.04(stat) and is consistent with no CP violation. The lifetime fit to the B⁰ → D*+D*- is used to verify the precision of the vertex resolution function. The measured lifetime is consistent with the world average value. Finally, the CP study of the B⁰ → D*+D*- decay gives an evidence of CP violation with a 3.5 σ significance: where S'D*+D*- = 1/η SD*+D*-. This result is consistent with the SM prediction for tree decays.