We report the results of a search for the anomalous production of a massive particle decaying to four electrons via two Z 0 bosons in 1:1 fb À1 of p " p collisions at ffiffi ffi s p ¼ 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=c 2 to be 0:028 AE 0:009ðstatÞ AE 0:011ðsystÞ events. We observe zero events in this search region. Assuming a Randall-Sundrum graviton production model, we set 95% C.L. limits on  BFðG ! Z 0 Z 0 Þ < 4-6 pb, depending on the graviton mass.