We report the results of a study of neutral B meson decays to the D0π+π- final state, where the D0 is fully reconstructed. The results are obtained from an event sample containing 388×106 BB̄-meson pairs collected in the Belle experiment at the KEKB e+e- collider. The total branching fraction of the three-body decay B(B̄0→D0π+π-)=(8.4±0. 4(stat)±0.8(syst))×10-4 has been measured. The intermediate resonant structure of these three-body decays has been studied. From a Dalitz plot analysis we have obtained the product of the branching fractions for D2*+ and D0*+ production: B(B̄0→D2*+π-) ×B(D2*+→D0π+)=(2.15±0.17(stat)±0.29(syst) ±0.12(mod))×10-4, and B(B̄0→D0*+π-) ×B(D0*+→D0π+)=(0.60±0.13(stat)±0.15(syst) ±0.22(mod))×10-4. This is the first observation of the B̄0→D0*+π- decay. The B̄0→D0ρ0 and D0f2 branching fractions are measured to be: B(B̄0→D0ρ0)=(3. 19±0.20(stat)±0.24(syst)±0.38(mod))×10-4, and B(B̄0→D0f2)=(1.20±0.18(stat)±0.21(syst)±0.32(mod) )×10-4. © 2007 The American Physical Society.