The recent discovery of a large CP violating asymmetry in K(L) --> PI+ PI- E+
E- mode has prompted us to seach for the associated K(L) --> PI0 PI0 E+ E-
decay mode in the KTeV-E799 experiment at Fermilab. In 2.7E+11 K(L) decays, one
candidate event has been observed with an expected background of 0.3 event,
resulting in an upper limit for the K(L) --> PI0 PI0 E+ E- branching ratio of
6.6E-09 at the 90% confidence level.Comment: To be published in Phys. Rev. Let