We use K(L)'s in the 100-200 GeV energy range to produce 147 candidate events of the axial vector pair K1(1270)-K1(1400) in the nuclear Coulomb field of a Pb target and determine the radiative widths Gamma(K1(1400)-->K0+gamma)=280.8+/-23.2(stat)+/-40.4(syst) keV and Gamma(K1(1270)-->K0+gamma)=73.2+/-6.1(stat)+/-28.3(syst) keV. These first measurements appear to be lower than the quark-model predictions. We also place upper limits on the radiative widths for K(*)(1410) and K(*)(2)(1430) and find that the latter is vanishingly small in accord with SU(3) invariance in the naive quark model.