  We present the first measurement of the mass function of galaxy clusters
based directly on cluster masses derived from observations of weak
gravitational lensing. To investigate the degree of sample incompleteness
resulting from the X-ray based selection of the target clusters, we use a
sample of 50 clusters with weak lensing mass measurements to empirically
determine the relation between lensing mass and X-ray luminosity and the
scatter about this relation. We use a complete, volume-limited sub-sample of 35
X-ray luminous clusters of galaxies at 0.15<z<0.3 to constrain the abundance of
very massive (M >~ 10^15/h M_sun) clusters. From this, we constrain
sigma_8(Omega_m/0.3)^{0.37} = 0.67^{+0.04}_{-0.05} (68% confidence limits),
agreeing well with constraints from the 3-year WMAP CMB measurements and
estimates of cluster abundances based on X-ray observations, but somewhat lower
than constraints from ``cosmic shear'' weak lensing measurements in random
fields.
