We present a study of the decay B-→D(CP)0K- and its charge conjugate, where D(CP)0 is reconstructed in both a non-CP flavor eigenstate and in CP (CP-even and CP-odd) eigenstates, based on a sample of 382 million Υ(4S)→BB̅ decays collected with the BABAR detector at the PEP-II e+e- storage ring. We measure the direct CP asymmetries ACP± and the ratios of the branching fractions RCP±: ACP+=0.27±0.09(stat)±0.04(syst), ACP-=-0.09±0.09(stat)±0.02(syst), RCP+=1.06±0.10(stat)±0.05(syst), RCP-=1.03±0.10(stat)±0.05(syst). We also express the results in terms of the so-called Cartesian coordinates x+, x-, and r2: x+=-0.09±0.05(stat)±0.02(syst), x-=0.10±0.05(stat)±0.03(syst), r2=0.05±0.07(stat)±0.03(syst). These results will help to better constrain the phase parameter γ=arg(-VudVub*/VcdVcb*) of the Cabibbo-Kobayashi-Maskawa quark mixing matrix.