  We calculate the next-to-next-to-next-to-leading-order (N$^3$LO) QCD
corrections to inclusive vector-boson fusion (VBF) Higgs production at proton
colliders, in the limit in which there is no colour exchange between the
hadronic systems associated with the two colliding protons. We also provide
differential cross sections for the Higgs transverse momentum and rapidity
distributions. We find that the corrections are at the 1-2 permille level, well
within the scale-uncertainty of the next-to-next-to-leading-order (NNLO)
calculation. The associated scale-uncertainty of the N$^3$LO calculation is
typically found to be below the 2 permille level. We also consider theoretical
uncertainties due to missing higher order parton distribution functions, and
provide an estimate of their importance.
