The lifetimes of the $B^+$ and $B^0$ mesons have been measured using 
fully reconstructed decays.  In a sample of ~49600 J/psi -> mu+ mu- 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 distributions
of these B mesons give tau+ = 1.61 +- 0.16 +- 0.05 ps, 
tau0 = 1.57 +- 0.18 +- 0.08 ps, and tau+/tau0 = 1.02 +- 0.16 +- 0.05.