We have made high statistics measurements of tritium beta decay to set
a limit on the mass of the election antineutrino. The measurements us
ed a windowless gaseous source of molecular tritium and a high resolut
ion toroidal field magnetic spectrometer. We calibrate the system usin
g conversion electrons from the decay of Kr-83m. The best fit value fo
r the neutrino mass squared is significantly negative. An anomalous st
ructure in the beta decay spectrum in the last 55 eV closest to the en
d point causes this result.