  We update our estimate of the cross section for Higgs production in gluon
fusion at next-to-next-to-next-to-leading order (N$^3$LO) in $\alpha_s$ in view
of the recent full computation of the result in the soft limit for infinite top
mass, which determines a previously unknown constant. We briefly discuss the
phenomenological implications. Results are available through the updated
version of the ggHiggs code.
