  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 Br(B+
--> K1(1270)+ gamma) = (4.28 +- 0.94(stat.) +- 0.43(syst.)) x 10-5 with a
statistical significance of 9.2sigma. We find no significant signal for B+ -->
K1(1400)+ gamma and set an upper limit Br(B+ --> K1(1400)+ gamma) < 1.44 x 10-5
at the 90% confidence level. We also measure inclusive branching fractions for
B+ --> K+ pi+ pi- gamma and B0 --> K0 pi+ pi- gamma in the mass range 1 GeV/c^2
< M(K pi pi) < 2 GeV/c^2.
