  We present new results for the jet-veto efficiency and zero-jet cross section
in Higgs production through gluon fusion. We incorporate the N$^3$LO
corrections to the total cross section, the NNLO corrections to the 1-jet rate,
NNLL resummation for the jet $p_t$ and LL resummation for the jet radius
dependence. Our results include known finite-mass corrections and are obtained
using the jet-veto efficiency method, updated relative to earlier work to take
into account what has been learnt from the new precision calculations that we
include. For 13 TeV collisions and using our default choice for the
renormalisation and factorisation scales, $\mu_0=m_H/2$, the matched prediction
for the jet-veto efficiency increases the pure NNNLO prediction by about 2% and
the two have comparable uncertainties. Relative to NNLO+NNLL results, the new
prediction is 2% smaller and the uncertainty reduces from more than 10% to less
than 5%. Results are also presented for the central scale $\mu_0=m_H$.
