The main uncertainty in current determinations of the power spectrum normalization, sigma 8, from abundances of X-ray-luminous galaxy clusters arises from the calibration of the mass-temperature relation. We use our weak-lensing mass determinations of 30 clusters from the hitherto largest sample of clusters with lensing masses, combined with X-ray temperature data from the literature, to calibrate the normalization of this relation at a temperature of 8 keV, unk = (8.7 plus or minus 1.6) h super(-1) 10 super(14) M unk. This normalization is consistent with previous lensing-based results based on smaller cluster samples, and with some predictions from numerical simulations, but higher than most normalizations based on X-ray-derived cluster masses. Assuming the theoretically expected slope alpha = 3/2 of the mass-temperature relation, we derive sigma 8 = 0.88 plus or minus 0.09 for a spatially flat ACDM universe with Omega sub(w) = 0.3, The main systematic errors on the lensing masses result from extrapolating the cluster masses beyond the field of view used for the gravitational lensing measurements, and from the separation of cluster/background galaxies, contributing each with a scatter of 20%. Taking this into account, there is still significant intrinsic scatter in the mass-temperature relation indicating that this relation may not be very tight, at least at the high-mass end. Furthermore, we find that dynamically relaxed clusters are (75 plus or minus 40)% hotter than nonrelaxed clusters.