We have made high statistics measurements of tritium beta decay to set a limit on the mass of the electron antineutrino. The measurements used a windowless gaseous source of molecular tritium and a high resolution toroidal field magnetic spectrometer. We calibrate the system using conversion electrons from the decay of ${}^{83m}\mathrm{Kr}$. The best fit value for the neutrino mass squared is significantly negative. An anomalous structure in the beta decay spectrum in the last 55 eV closest to the end point causes this result.