The high precision determination of the partial width $Γ(H \longrightarrow γγ)$ 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 $γγ\longrightarrow q \bar{q}$, $q=\{b,c\}$, 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 $γγ\longrightarrow q \bar{q}$ are now known. We are therefore able to present for the first time precise predictions for the measurement of the partial width $Γ(H \longrightarrow γγ)$ at the Compton collider ($γγ$) 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 $\tan β$ impossible to cover at the LHC proton-proton collider.

25 pages, LATEX2e, eps-environment