The population of rich galaxy clusters evolves much more rapidly in a universe with critical density than in a universe with low density. Thus, counts of clusters at intermediate redshift offer the possibility of determining the cosmological density parameter, Ω0, with a minimum of assumptions. We quantify this evolution using the Press-Schechter formalism which we extend to flat cosmological models with a cosmological constant, Λ0 = 1 − Ω0 Using new large N-body simulations, we verify that this formalism accurately predicts the abundance of rich clusters as a function of redshift in various cosmologies. We normalize the models by comparing them with the local abundance of clusters as a function of their X-ray temperature which we rederive from data compiled by Henry & Arnaud. The resulting values of the rms density fluctuation in spheres of radius 8h−1 Mpc are σ8 = (0.52 ± 0.04)Ω0−0.46+0.10Ω0 if Λ0 = 0 and σ8 = (0.52±0.04)Ω0−0.52+0.13Ω0 if Λ0 = 1−Ω0. These values depend only weakly, and almost not at all if ォ0 = 1, on the shape of the power spectrum. We then examine how the distributions of mass, X-ray temperature and Sunyaev-Zel'dovich decrement evolve as a function of ォ0. We present the expected distributions at z = 0.33 and z = 0.5 and the predicted number counts of the largest clusters, both in space and in projection on the sky. We find that, even at z = 0.33, these distributions depend very strongly on ォ0 and only weakly on ゛0. For example, at this redshift, we expect 15 times as many clusters per comoving volume with if > 3.5 × 1010h−1 M⊙ and 5 times as many clusters with kT > 5 keV if Ω0 = 0.3 than if Ω0 = 1. The splitting in the integrated counts is enhanced by the larger volume element in low-Ω0 models. There is therefore a real prospect of estimating Ω0 from forthcoming surveys of intermediate-redshift clusters that will determine their masses, X-ray temperatures or Sunyaev-Zel'dovich decrements.