  The number density of rich galaxy clusters still provides the most robust way
of normalizing the power spectrum of dark matter perturbations on scales
relevant to large-scale structure. We revisit this constraint in light of
several recent developments: (1) the availability of well-defined samples of
local clusters with relatively accurate X-ray temperatures; (2) new theoretical
mass functions for dark matter haloes which provide a good fit to large
numerical simulations; (3) more accurate mass-temperature relations from larger
catalogs of hydrodynamical simulations; (4) the requirement to consider closed
as well as open and flat cosmologies to obtain full multi-parameter likelihood
constraints for CMB and SNe studies. We present a new sample of clusters drawn
from the literature and use this sample to obtain improved results on sigma_8,
the normalization of the matter power spectrum on scales of 8 h^{-1} Mpc, as a
function of the matter density and cosmological constant in a Universe with
general curvature. We discuss our differences with previous work, and the
remaining major sources of uncertainty. Final results on the 68 per cent
confidence region, approximately independent of power spectrum shape, can be
expressed as constraints on sigma at an appropriate cluster normalization scale
R_Cl. We provide fitting formulas for R_Cl and sigma(R_Cl) for general
cosmologies, as well as for sigma_8 as a function of cosmology and shape
parameter Gamma. For flat models we find approximately sigma_8 \simeq
0.495^{+0.034}_{-0.037}) Omega_M^{-0.60} for Gamma=0.23, where the error bar is
dominated by uncertainty in the mass-temperature relation.
