We present for the first time a detailed and comprehensive analysis of the experimental results that set the current world sensitivity limit on the magnitude of the electric dipole moment (EDM) of the neutron. We have extended and enhanced our earlier analysis to include recent developments in the understanding of the effects of gravity in depolarizing ultracold neutrons (UCN); an improved calculation of the spectrum of the neutrons; and conservative estimates of other possible systematic errors, which are also shown to be consistent with more recent measurements undertaken with the apparatus. We obtain a net result of $d_mathrm{n} = -0.21 pm 1.82 times10^{-26}$ $e$cm, which may be interpreted as a slightly revised upper limit on the magnitude of the EDM of $3.0 times10^{-26}$ $e$cm (90% CL) or $ 3.6 times10^{-26}$ $e$cm (95% CL). This paper is dedicated by the remaining authors to the memory of Prof. J. Michael Pendlebury.
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