  We present the inclusive cross section at next-to-next-to-next-to-leading
order (N$^3$LO) in perturbative QCD for the production of a Higgs boson via
bottom-quark fusion. We employ the five-flavour scheme, treating the bottom
quark as a massless parton while retaining a non-vanishing Yukawa coupling to
the Higgs boson. We find that the dependence of the hadronic cross section on
the renormalisation and factorisation scales is substantially reduced. For
judicious choices of the scales the perturbative expansion of the cross section
shows a convergent behaviour. We present results for the N$^3$LO cross section
at various collider energies. In comparison to the cross section obtained from
the Santander-matching of the four and five-flavour schemes we predict a
slightly higher cross section, though the two predictions are consistent within
theoretical uncertainties.
