  Using 9.6 million B meson pairs collected with the CLEO detector, we have
fully reconstructed 135 B0 -> psi(') K_S and 526 B+ -> psi(') K+ candidates
with very low background.
  We fitted the psi(')K invariant mass distributions of these B meson
candidates and measured the masses of the neutral and charged B mesons to be
M(B0)=5279.1+-0.7[stat]+-0.3[syst] MeV/c^2 and
M(B+)=5279.1+-0.4[stat]+-0.4[syst] MeV/c^2. The precision is a significant
improvement over previous measurements.
