  We present one-loop contributions to the fully differential Higgs boson
gluon-fusion cross-section for Higgs production via gluon fusion. Our results
constitute a necessary ingredient of a complete N$^3$LO determination of the
cross-section. We perform our computation using a subtraction method for the
treatment of soft and collinear singularities. We identify the infrared
divergent parts in terms of universal splitting and eikonal functions, and
demonstrate how phase-space integrations yield poles (up to $1/\epsilon^6$ ) in
the dimensional regulator $\epsilon=(4-d)/2$. We compute the coefficients of
the $\epsilon$ expansion, including the finite part numerically. As a
demonstration of our numerical implementation, we present the corrections at
N$^3$LO due to one-loop amplitudes in the rapidity and transverse momentum of
the Higgs boson.
