  We describe the computation of the $gg \to W^+W^-g$ process that contributes
to the production of two $W$-bosons and a jet at the CERN Large Hadron Collider
(LHC). While formally of next-to-next-to-leading order (NNLO) in QCD, this
process can be evaluated separately from the bulk of NNLO QCD corrections
because it is finite and gauge-invariant. It is also enhanced by the large
gluon flux and by selection cuts employed in the Higgs boson searches in the
decay channel $ H \to W^+W^-$, as was first pointed out by Binoth {\it et al.}
in the context of $gg \to W^+W^-$ production. For cuts employed by the ATLAS
collaboration, we find that the gluon fusion contribution to $pp \to W^+W^-j$
enhances the background by about ten percent and can lead to moderate
distortions of kinematic distributions which are instrumental for the ongoing
Higgs boson searches at the LHC. We also release a public code to compute the
NLO QCD corrections to this process, in the form of an add-on to the package
{\tt MCFM}.
