We report the observation of charmless hadronic decays of charged B mesons to the final state K{+}K{-}pi{+}. Using a data sample of 347.5 fb{-1} collected at the Upsilon(4S) resonance with the BABAR detector, we observe 429+/-43 signal events with a significance of 9.6sigma. We measure the inclusive branching fraction B(B{+}-->K{+}K{-}pi{+})=[5.0+/-0.5(stat)+/-0.5(syst)]x10{-6}. Inspection of the Dalitz plot of signal candidates shows a broad structure peaking near 1.5 GeV/c{2} in the K+K- invariant mass distribution. We find the direct CP asymmetry to be consistent with zero.