  The high precision determination of the partial width $\Gamma (H
\longrightarrow \gamma \gamma)$ of an intermediate mass Higgs boson is among
the most important measurements at a future photon--photon collider. Recently
it was shown that large non-Sudakov as well as Sudakov double logarithmic
corrections can be summed to all orders in the background process $\gamma
\gamma \longrightarrow q \bar{q}$, $q=\{b,c\}$, from an initially polarized
$J_z=0$ state. In addition, running coupling corrections were included exactly
to all orders by employing the renormalization group. Thus all necessary
theoretical results for calculating the Higgs signal and the non-Higgs
continuum background contributions to the process $\gamma \gamma
\longrightarrow q \bar{q}$ are now known. We are therefore able to present for
the first time precise predictions for the measurement of the partial width
$\Gamma (H \longrightarrow \gamma \gamma)$ at the Compton collider
($\gamma\gamma$) option at a future linear $e^+e^-$ collider. The interplay
between signal and background is very sensitive to the experimental cuts and
the ability of the detectors to identify $b$-quarks in the final state. We
investigate this in some detail using a Monte Carlo analysis, and conclude that
a measurement with a 2 % statistical accuracy should be achievable. This could
have important consequences for the discovery of physics beyond the Standard
Model, in particular for large masses of a pseudoscalar Higgs boson as the
decoupling limit is difficult and for a wide range of $\tan \beta$ impossible
to cover at the LHC proton-proton collider.
