This work presents a precision determination of the mass difference of neutral and charged B mesons, m(B0) - m(B+). The measurement is based on a data sample of about 232 million B-meson pairs recorded with the BABAR detector at the e+e- storage-ring system PEP-II. In events of e+e- --> Upsilon(4S) --> B Bbar, B0 and B+ mesons are fully reconstructed in decays B+ --> J/psi K+ and B0 --> J/psi K*0 using the subsequent decays J/psi --> l+ l- and K*0 --> K+ pi- for the reconstruction of J/psi mesons and K*0 resonances, respectively. The determination of the mass difference from the measurement of the invariant B-meson masses suffers from detector-resolution effects. To reach the desired sensitivity, the B-meson momenta are measured in the center-of-mass system, the Upsilon(4S) rest frame. Since their energy is also known in the center-of-mass system, this method leads to the mass difference using the relativistic energy-momentum relation. For this purpose, three different fit methods are performed to the spectra of the B momenta. The obtained result is m(B0) - m(B+) = (+0.33 +- 0.05 +- 0.03) MeV/c^2, where the first error is statistical and the second estimates the systematic uncertainty of this measurement, which agrees well with the current world average. However, the combined error of this measurement is substantially smaller than that in the current world average, and the significance of m(B0) - m(B+) being non-zero has exceeded the 5 sigma level.