  We compute the NLO QCD corrections to the loop-induced gluon fusion
contribution in $W^+W^-$ production at the LHC. We consider the full leptonic
process $pp\to \ell^+\ell^{\prime\, -}\nu_{\ell}{\bar\nu}_{\ell^\prime}+X$, by
including resonant and non-resonant diagrams, spin correlations and off-shell
effects. Quark-gluon partonic channels are included for the first time in the
calculation, and our results are combined with NNLO predictions to the quark
annihilation channel at the fully differential level. The computed corrections,
which are formally of ${\cal O}(\alpha_{\mathrm{S}}^3)$, increase the NNLO
cross section by only about 2%, but have an impact on the shapes of kinematical
distributions, in part due to the jet veto, which is usually applied to reduce
the top-quark background. Our results, supplemented with NLO EW effects,
provide the most advanced fixed-order predictions available to date for this
process, and are compared with differential ATLAS data at $\sqrt{s}=$ 13 TeV.
