We present the results of a search for the rare flavor-changing neutral-current decays B ! ' þ ' À ( ¼ AE , 0 and ' ¼ e, ) and B 0 ! ' þ ' À using a sample of e þ e À ! Çð4SÞ ! B "B decays corresponding to 428 fb À1 of integrated luminosity collected by the BABAR detector. No significant signal is observed, and we set an upper limit on the isospin and lepton-flavor averaged branching fraction of BðB ! ' þ ' À Þ < 5:9 Â 10 À8 and a lepton-flavor averaged upper limit of BðB 0 ! ' þ ' À Þ < 6:4 Â 10 À8 , both at the 90% confidence level. We also report 90% confidence level branching fraction upper limits for the individual modes B þ ! þ e þ e À , B 0 ! 0 e þ e À , B þ ! þ þ À , B 0 ! 0 þ À , B 0 ! e þ e À , and B 0 ! þ À .