Using 449x10(6) BB[over ] pairs collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider, we observe clear signals for B(+)-->K[over ](0)K(+) and B(0)-->K[over ](0)K(0) decays with 5.3sigma and 6.0sigma significance, respectively. We measure the branching fractions B(B(+)-->K[over ](0)K(+))=(1.22(-0.28-0.16)(+0.32+0.13))x10(-6) and B(B(0)-->K[over ](0)K(0))=(0.87(-0.20)(+0.25)+/-0.09)x10(-6), and partial-rate asymmetries A(CP)(B(+)-->K[over ](0)K(+))=0.13(-0.24)(+0.23)+/-0.02 and A(CP)(B(0)-->K[over ](0)K(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.