  The impact of the finite top-quark mass on the inclusive Higgs production
cross section at higher perturbative orders has been an open question for
almost three decades. In this paper, we report on the computation of this
effect at NNLO QCD. For the purely gluonic channel, it amounts to $+0.62\%$
relative to the result obtained in the HEFT approximation. The formally
sub-leading partonic channels over-compensate this shift, leading to an overall
effect of $-0.32\%$ at a $pp$ collider energy of $13\,\mathrm{TeV}$, and
$-0.16\%$ at $8\,\mathrm{TeV}$. This result eliminates one of the main
theoretical uncertainties to inclusive Higgs production cross section at the
LHC.
