  The first four terms of an expansion in mH^2/mt^2 of the total inclusive
cross section for Higgs production in gluon fusion are evaluated through
next-to-next-to-leading order QCD. A reliable and precise approximation of the
full top mass dependence at NNLO is derived and compared to the frequently used
heavy-top limit. It is found that both results agree numerically to better than
0.5% in the Higgs mass range of 100-300 GeV. This validates the higher order
results for the inclusive Higgs cross section and justifies the heavy-top limit
as a powerful tool for Higgs phenomenology at the LHC and the Tevatron.
