We report measurements of B meson decays to two kaons using 253 fb(-1) of data collected with the Belle detector at the KEKB energy-asymmetric e+ e- collider. We find evidence for signals in B+ --> K0 K+ and B0 --> K0 K0 with significances of 3.0sigma and 3.5sigma, respectively. (Charge-conjugate modes are included.) The corresponding branching fractions are measured to be [symbol: see text](B+ --> K0 K+) = (1.0 +/- 0.4 +/- 0.1) x 10(-6) and [symbol: see text](B0 --> K0 K0) = (0.8 +/- 0.3 +/- 0.1) x 10(-6). These decay modes are examples of hadronic bd transitions. No signal is observed in the decay B0 --> K+ K-, and we set an upper limit of 3.7 x 10(-7) at 90% confidence level.