  We report on a calculation of the cross-section for Higgs boson production in
gluon fusion in association with a hadronic jet at next-to-next-to-leading
order (NNLO) in perturbative QCD. The computational technique is discussed in
detail. We show explicitly how to employ known soft and collinear limits of
scattering amplitudes to construct subtraction terms for NNLO computations.
Cancellation of singularities is demonstrated numerically for the
collinearly-subtracted $gg \to H + j $ cross-section through NNLO and the
finite $\sigma_{gg \to Hj}$ cross-section is computed through ${\cal
O}(\alpha_s^5)$ as a function of the center-of-mass collision energy. We
present numerical results for the gluon-fusion contribution to Higgs production
in association with a jet at the LHC. The NNLO QCD corrections significantly
reduce the residual scale dependence of the cross-section. The computational
method that we describe in this paper is applicable to the calculation of NNLO
QCD corrections to any other $ 2 \to 2$ process at a hadron collider
