  We report studies of the decay B+- -> D_CP K+-, where D_CP denotes neutral D
mesons that decay to CP eigenstates. The analysis is based on a 29.1/fb data
sample of collected at the \Upsilon(4S) resonance with the Belle detector at
the KEKB asymmetric e+ e- storage ring. Ratios of branching fractions of
Cabibbo-suppressed to Cabibbo-favored processes involving D_CP are determined
to be B(B- -> D_1 K-)/B(B- -> D_1 pi-)=0.125 +- 0.036 +- 0.010 and B(B- -> D_2
K-)/B(B- -> D_2 pi-)=0.119 +- 0.028 +- 0.006, where indices 1 and 2 represent
the CP=+1 and CP=-1 eigenstates of the D0 - anti D0 system, respectively. We
also extract the partial rate asymmetries for B+- -> D_CP K+-, finding A_1 =
0.29 +- 0.26 +- 0.05 and A_2 = -0.22 +- 0.24 +- 0.04.
