We report the observation of the radiative decay B+-->K1(1270)(+) gamma using a data sample of 140 fb(-1) taken at the Upsilon(4S) resonance with the Belle detector at the KEKB e+e- collider. We find the branching fraction to be B(B+-->K1(1270)(+)gamma)=(4.3+/-0.9(stat.)+/-0.9(syst.))x10(-5) with a significance of 7.3sigma. We find no significant signal for B+-->K1(1400)(+)gamma and set an upper limit B(B+-->K1(1400)(+)gamma)<1.5 x 10(-5) at the 90% confidence level. We also measure inclusive branching fractions for B+-->K+pi+pi-gamma and B0-->K0pi+pi-gamma in the mass range 1 GeV/c(2)<M(K+(0)pi+pi-)<2 GeV/c(2).