We have obtained precise measurements of the Sigma(0) mass and the Sig
ma(0)-Lambda(0) mass difference from a fit to the Lambda(0) gamma inva
riant mass distribution of 3327 Sigma(0)-->Lambda(0)+gamma decays. Our
measurements yield M-Sigma 0 = 1192.65 +/- 0.020 +/- 0.014 MeV/c(2) a
nd M-Sigma 0-M-Lambda 0 = 76.966 +/- 0.020 +/- 0.013 MeV/c(2), where t
he uncertainties are statistical and systematic in that order. This re
presents a significant improvement over all previous determinations an
d is the first direct measurement of the Sigma(0) mass itself.