Using 449x10(6) B(B) over bar pairs collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider, we observe clear signals for B+->(K) over bar K-0(+) and B-0 ->(K) over bar K-0(0) decays with 5.3 sigma and 6.0 sigma significance, respectively. We measure the branching fractions B(B+->(K) over bar K-0(+))=(1.22(-0.28-0.16)(+0.32+0.13))x10(-6) and B(B-0 ->(K) over bar K-0(0))=(0.87(-0.20)(+0.25)+/- 0.09)x10(-6), and partial-rate asymmetries A(CP)(B+->(K) over bar K-0(+))=0.13(-0.24)(+0.23)+/- 0.02 and A(CP)(B-0 ->(K) over bar K-0(0))=-0.58(-0.66)(+0.73)+/- 0.04. From a simultaneous fit, we also obtain B(B+-> K-0 pi(+))=(22.8(-0.7)(+0.8)+/- 1.3)x10(-6) and A(CP)(B+-> K-0 pi(+))=0.03 +/- 0.03 +/- 0.01. The first and second error in the branching fractions and the partial-rate asymmetries are statistical and systematic, respectively. No signal is observed for B-0 -> K+K- decays, and for this branching fraction, we set an upper limit of 4.1x10(-7) at the 90% confidence level.