The paper reports on the improved Mainz experiment on tritum $beta$ spectroscopy which yields a 10 times higher signal to background ratio than before. The main experimental effects and systematic uncertainties have been investigated in side experiments and possible error sources have been eliminated. Extensive data taking took place in the years 1997 to 2001. A residual analysis of the data sets yields for the square of the electron antineutrino mass the final result of $m^2( u_e)=(-0.6 pm 2.2_{rm{stat}} pm 2.1_{rm{syst}})$ eV$^2$/c$^4$. We derive an upper limit of $m( u_e)leq 2.3$ eV/c$^2$ at 95% confidence level for the mass itself.
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