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We forecast constraints on the amplitude of matter clustering sigma_8(z) achievable with the combination of cluster weak lensing and number counts, in current and next-generation weak lensing surveys. We advocate an approach, analogous to galaxy-galaxy lensing, in which the observables in each redshift bin are the mean number counts and the mean weak lensing profile of clusters above a mass proxy threshold. The primary astrophysical nuisance parameter is the logarithmic scatter between the mass proxy and true mass near the threshold. For surveys similar to the Dark Energy Survey (DES), the Roman Space Telescope High Latitude Survey (HLS), and the Rubin Observatory Legacy Survey of Space and Time (LSST), we forecast aggregate precision on sigma_8 of 0.26%, 0.24%, and 0.10%, respectively, if the mass-observable scatter has an external prior better than 0.01. These constraints would be degraded by about 20% for a 0.05 prior on scatter in the case of DES or HLS and for a 0.016 prior for LSST. A one-month observing program with Roman Space Telescope targeting approximately 2500 massive clusters could achieve a 0.5% constraint on sigma_8(z=0.7) on its own, or a ~0.33% constraint in combination with the HLS. Realizing the constraining power of clusters requires accurate knowledge of the mass-observable relation and stringent control of systematics. We provide analytic approximations to our numerical results that allow easy scaling to other survey assumptions or other methods of cluster mass estimation.