We report the most sensitive direct upper limit set on the mass ${\mathit{m}}_{\ensuremath{\nu}}$ of the electron antineutrino. Our measurements of the shape of the \ensuremath{\beta} decay spectrum of free molecular tritium yield, under the assumption of no new physics other than that of mass, a central value for ${\mathit{m}}_{\ensuremath{\nu}}^{2}$ of -147\ifmmode\pm\else\textpm\fi{}68\ifmmode\pm\else\textpm\fi{}41 ${\mathrm{eV}}^{2}$, which corresponds to an upper limit of 9.3 eV (95% confidence level) on ${\mathit{m}}_{\ensuremath{\nu}}$. The result is in clear disagreement with a reported value of 26(5) eV.