We report the results of a search for the anomalous production of a massive particle decaying to four electrons via two Z0 bosons in 1.1  fb−1 of pp collisions at s√=1.96  TeV collected by the CDF II detector at Fermilab. We employ optimized electron identification criteria to maximize acceptance and efficiency. We estimate the backgrounds in the invariant mass range 500–1000  GeV/c2 to be 0.028±0.009(stat)±0.011(syst) events. We observe zero events in this search region. Assuming a Randall-Sundrum graviton production model, we set 95% C.L. limits on σ×BF(G→Z0Z0)<4–6  pb, depending on the graviton mass.