We present measurements of the masses of 20 X-ray luminous clusters of galaxies at intermediate redshifts, determined from a weak lensing analysis of deep archival R-band data obtained using the Canada-France-Hawaii-Telescope. Compared to previous work, our analysis accounts for a number of effects that are typically ignored, but can lead to small biases, or incorrect error estimates. We derive masses that are essentially model independent and find that they agree well with measurements of the velocity dispersion of cluster galaxies and with the results of X-ray studies. Assuming a power law between the lensing mass and the X-ray temperature, M_2500 T^alpha, we find a best fit slope of alpha=1.34^{+0.30}_{-0.28}. This slope agrees with self-similar cluster models and studies based on X-ray data alone. For a cluster with a temperature of kT=5keV we obtain a mass M_{2500}=(1.4+-0.2)\times 10^{14}h^{-1}Msun in fair agreement with recent Chandra and XMM studies.