Using the entire sample of 467×10^6 Υ(4S)→BB[overbar] decays collected with the BABAR detector at the PEP-II asymmetric-energy B factory at the SLAC National Accelerator Laboratory, we perform an analysis of B^± → DK^± decays, using decay modes in which the neutral D meson decays to either CP-eigenstates or non-CP-eigenstates. We measure the partial decay rate charge asymmetries for CP-even and CP-odd D final states to be A_(CP+) = 0.25±0.06±0.02 and A_(CP-) = -0.09±0.07±0.02, respectively, where the first error is the statistical and the second is the systematic uncertainty. The parameter A_(CP+) is different from zero with a significance of 3.6 standard deviations, constituting evidence for direct CP violation. We also measure the ratios of the charged-averaged B partial decay rates in CP and non-CP decays, R_(CP+) = 1.18±0.09±0.05 and R_(CP-) = 1.07±0.08±0.04. We infer frequentist confidence intervals for the angle γ of the unitarity triangle, for the strong phase difference δ_B, and for the amplitude ratio r_B, which are related to the B^- → DK^- decay amplitude by r_(B)e^(i(δB-γ)) = A(B^- → D[overbar]^(0)K^-)/A(B^- → D^(0)K^-). Including statistical and systematic uncertainties, we obtain 0.24 < r_B < 0.45 (0.06 < r_B <0.51) and, modulo 180°, 11.3° < γ < 22.7° or 80.8° < γ <99.2° or 157.3° <γ < 168.7° (7.0°<γ<173.0°) at the 68% (95%) confidence level
