Using 449×106 BB̄ pairs collected with the Belle detector at the KEKB asymmetric-energy e+e- collider, we observe clear signals for B+→K̄0K+ and B0→K̄0K0 decays with 5.3σ and 6.0σ significance, respectively. We measure the branching fractions B(B+→K̄0K+)=(1.22-0.28-0.16+0.32+0.13)×10-6 and B(B0→K̄0K0)=(0.87-0.20+0.25±0.09)×10-6, and partial-rate asymmetries ACP(B+→K̄0K+)=0.13-0.24+0.23±0.02 and ACP(B0→K̄0K0)=-0.58-0.66+0.73±0.04. From a simultaneous fit, we also obtain B(B+→K0π+)=(22.8-0.7+0.8±1.3)×10-6 and ACP(B+→K0π+)=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 B0→K+K- decays, and for this branching fraction, we set an upper limit of 4.1×10-7 at the 90% confidence level.