  We compute the O(alpha_s^2) QCD corrections to the fully differential
cross-section pp \to W X \to l \nu X, retaining all effects from spin
correlations. The knowledge of these corrections makes it possible to calculate
with high precision the W boson production rate and acceptance at the LHC,
subject to realistic cuts on the lepton and missing energy distributions. For
certain choices of cuts we find large corrections when going from
next-to-leading order (NLO) to next-to-next-to-leading order (NNLO) in
perturbation theory. These corrections are significantly larger than those
obtained by parton-shower event generators merged with NLO calculations. Our
calculation may be used to assess and significantly reduce the QCD
uncertainties in the many studies of $W$ boson production planned at the LHC.
