The lifetimes of B+ and B0 mesons are measured using a sample of 150 000 hadronic Z0 decays collected by the SLD experiment at the SLAC Linear Collider between 1993 and 1995. Two analyses are presented in which the decay length and charge of the B meson are reconstructed. The first method uses a novel topological vertexing technique while the second uses semi-inclusively reconstructed semileptonic decays. The topological analysis yields a sample of 6033 (3665) charged (neutral) vertices with good charge purity, whereas the semileptonic analysis yields a smaller sample of 634 (584) charged (neutral) decays with excellent charge purity. Combining the results from both analyses, we find τB+ = 1.66±0.06(stat)±0.05(syst) ps, τB0 = 1.64±0.08(stat)±0.08(syst) ps, and τB+/τB0 = 1.01±0.07(stat)±0.06(syst).