The final results of the ALEPH search for the Standard Model Higgs boson at LEP, with data collected in the year 2000 at centre-of-mass energies up to 209,GeV, are presented. The changes with respect to the preceding publication are described and a complete study of systematic effects is reported. The findings of this final analysis confirm the preliminary results published in November 2000 shortly after the closing down of the LEP collider: a significant excess of events is observed, consistent with the production of a $115, Gcs$ Standard Model Higgs boson.% The final results of the searches for the neutral Higgs bosons of the MSSM are also reported, in terms of limits on $mh$, $mA$ and $anb$.% Limits are also set on $mh$ in the case of invisible decays.

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