The lifetimes of the B+ and B0 mesons have been measured using fully r
econstructed decays. In a sample of approximately 49 600 J/psi --> mumu- decays recorded with the Collider Detector at Fermilab, 148 +/- 16
B+ and 121 +/- 16 B0 mesons have been reconstructed using the silicon
vertex detector. Unbinned likelihood fits to the proper lifetime dist
ributions of these B mesons give tau+ = 1.61 +/- 0.16 (stat) +/- 0.05
(syst) ps, tau0 = 1.57 +/- 0.18 (stat) +/- 0.08 (syst) ps, and tau+/ta
u0 = 1.02 +/- 0.16 (stat) +/- 0.05 (syst).