  We present an estimate of the next-to-leading order QCD corrections to mixed
QCD-electroweak contribution to Higgs boson production cross section in gluon
fusion, combining the recently computed three-loop virtual corrections and the
approximate treatment of real emission in the soft approximation. We find that
the NLO QCD corrections to mixed QCD-electroweak contributions are nearly
identical to NLO QCD corrections to QCD Higgs production. Our result confirms
an earlier estimate of these ${\cal O}\left( \alpha\, \alpha_s^2 \right)$
effects in arXiv:0811.3458 [hep-ph] and provides further support for the
factorization approximation of QCD and electroweak corrections.
