We report on the first observation of the decay K-L --> pi (0)e(+)e(-)gamma by the KTeV E799 experiment at Fermilab. Based upon a sample of 48 events with an estimated background of 3.6 +/- 1.1 events, we measure the K-L --> pi (0)e(+)e(-)gamma branching ratio to be (2.34 +/- 0.35 +/- 0.13) x 10(-8). Our data agree with recent O(p(6)) calculations in chiral perturbation theory that include contributions from vector meson exchange through the parameter a(V). A fit was made to the K-L --> pi (0)e(+)e(-)gamma data for a(V) with the result -0.67 +/- 0.21 +/- 0.12, which is consistent with previous results from KTeV.