  We show that the counts of galaxy clusters in future deep cluster surveys can
place strong constraints on the matter density, Omega_m, the vacuum energy
density, Omega_L, and the normalization of the matter power spectrum, sigma_8.
Degeneracies between these parameters are different from those in studies of
either high--redshift type Ia Supernovae (SNe), or cosmic microwave background
(CMB) anisotropies. Using a mass threshold for cluster detection expected to be
typical for upcoming SZE surveys, we find that constraints on Omega_m and
sigma_8 at the level of roughly 5% or better can be expected, assuming redshift
information is known at least to z=0.5 and in the absence of significant
systematic errors. Without information past this redshift, Omega_L is
constrained to 25%. With complete redshift information, deep (M_{lim}=
10^{14}h^{-1}{M_sun}), relatively small solid angle (roughly 12 {deg}^2)
surveys can further constrain Omega_L to an accuracy of 15%, while large solid
angle surveys with ground-based large-format bolometer arrays could measure
Omega_L to a precision of 4% or better.
