  We resum a class of large Sudakov logarithms affecting Higgs boson production
in the exclusive one-jet bin at the LHC. We extend previous results by
calculating the full one-loop soft function for this process, which extends the
accuracy of the resummation to include the leading three logarithmic
corrections at each order in the QCD coupling constant. We match this result to
the next-to-leading order cross section and present a detailed numerical study
assuming realistic LHC cuts. Careful attention is paid to the matching
procedure, and to the theoretical uncertainties induced by residual scale
variation. We find that the matched NLL'+ NLO cross section has significantly
smaller uncertainties than the fixed-order result, and can be used to alleviate
the theoretical errors hindering current Higgs analyses at the LHC.
