We study the ability of weak lensing surveys to detect galaxy clusters and constrain cosmological parameters, in particular the equation of state of dark energy. There are two major sources of noise for weak lensing cluster measurements: the ``shape noise'' from the intrinsic ellipticities of galaxies; and the large scale projection noise. We produce a filter for the shear field which optimizes the signal-to-noise of shape-noise-dominated shear measurements. Our Fisher-matrix analysis of this projected-mass observable makes use of the shape of this mass function, and takes into account the Poisson variance, sample variance, shape noise, and projected-mass noise, and also the fact that the conversion of the shear signal into mass is cosmology-dependent. The Fisher analysis is applied to both a nominal 15,000 square degree ground-based survey and a 1000 square degree space-based survey. Assuming a detection threshold of S/N=5, we find both experiments detect \~20,000 clusters, and yield 1-sigma constraints of ~0.07 for w0 and ~0.2 for wa when combined with CMB data (for flat universe). The projection noise exceeds the shape noise only for clusters at z<=0.1 and has little effect on the derived dark-energy constraints. Sample variance does not significantly affect either survey. Finally, we note that all these results are extremely sensitive to the noise levels and detection thresholds that we impose. They can be significantly improved if we combine ground and space surveys as independent experiments and add their corresponding Fisher matrices. Comment: Accepted for publication in Physical Review D