We produce a sample consisting of 147 candidate events, with minimal backgrounds, of the mixed axial vector pair (K 1 (1270)-K 1 (1400)) by exciting K L 's in the Coulomb field of lead and report the first measurements of the radiative widths Γ r (K 1 (1400)) = 280.8 ± 23.2(stat) ± 40.4(syst) keV and Γ r (K 1 (1270)) = 73.2±6.1(stat)±28.3(syst) keV. We also place 90% CL upper limits Γ r (K * (1410)) ≤ 52.9 keV for the vector state and Γ r (K * 2 (1430)) ≤ 5.4 keV for the tensor state. These measurements allow for significant tests of quark-model predictions of radiative widths for the low-lying vector mesons.