We present the next-to-leading-order QCD corrections to the production of a Higgs boson in association with one jet at the LHC including the full top-quark mass dependence. The mass of the bottom quark is neglected. The two-loop integrals appearing in the virtual contribution are calculated numerically using the method of sector decomposition. We study the Higgs boson transverse momentum distribution, focusing on the high p_{t,H} region, where the top-quark loop is resolved. We find that the next-to-leading-order QCD corrections are large but that the ratio of the next-to-leading-order to leading-order result is similar to that obtained by computing in the limit of large top-quark mass.