  We present a next-to-leading order calculation of H+jet in gluon fusion
including the effect of a finite top quark mass $m_t$ at large transverse
momenta. Using the recently published two-loop amplitudes in the high energy
expansion and our previous setup that includes finite $m_t$ effects in a low
energy expansion, we are able to obtain $m_t$-finite results for transverse
momenta below 225 GeV and above 500 GeV with negligible remaining top quark
mass uncertainty. The only remaining region that has to rely on the common
leading order rescaling approach is the threshold region $\sqrt{\hat s}\simeq
2m_t$. We demonstrate that this rescaling provides an excellent approximation
in the high $p_T$ region. Our calculation settles the issue of top quark mass
effects at large transverse momenta. It is implemented in the parton level
Monte Carlo code MCFM and is publicly available immediately in version 8.2.
