We study the decays B- → D*0π - and B-→ D*0K-, where the D*0 decays into D0π0, with the D0 reconstructed in the CP-even (CP+) eigenstates K-K + and π-π+ and in the (non-CP) channels K-π+, K -π+π+π-, and K -π+π0. Using a sample of about 123 × 106 BB̄ pairs, we measure the ratios of decay rates R*non-CP ≡ B(B- → D* non-CP0K-)/B(B- → D*non-CP0π-) = 0.086 ± 0.0040(stat)-0.0031+0.0042(syst), and provide the first measurements of R*CP+ ≡ B(B- → D*CP+0K-)/B(B- → D*CP+0π-) = 0.0086 ± 0.021(stat) ± 0.007(syst), and of the CP asymmetry A*CP+ ≡ B(B- → D*CP+0K -)-B(B+ → D*CP+0K +)/B(B- → D*CP+0K -)+B(B+ → D*CP+0K +) = 0.10 ± 0.23(stat)-0.04+0.03(syst). © 2005 The American Physical Society.