  We compute the mixed QCD-electroweak corrections to the cross section for the
production of a Higgs boson via gluon fusion, in the limit of a small mass of
the electroweak gauge bosons. This limit is regular and we calculate it by
setting the W, Z masses to zero in the Feynman rules for their propagators. Our
analytic results provide an independent check, in a non-trivial limit, of a
recent exact computation for the three-loop mixed QCD and electroweak virtual
corrections and the corresponding contribution to the cross section in the
soft-virtual approximation. From our calculation in the small mass
approximation, we can infer the second term in the expansion of the cross
section around the threshold limit with its exact dependence on the masses of
the W, Z bosons. Furthermore we find that in the small mass approximation the
non-factorizable contributions from the real radiation, so far unknown for full
gauge boson mass dependence, are modest in comparison to the known factorizable
and virtual contributions to the full ${\cal O}(\alpha_s^3 \alpha^2)$ mixed QCD
and electroweak cross-section. This furnishes a new phenomenological test of
estimates for the mixed QCD and electroweak corrections, which were based on
the hypothesis of factorization of QCD and electroweak corrections.
