  The mean sky-averaged Comptonization parameter, y, describing the scattering
of the CMB by hot gas in clusters of galaxies is calculated in an array of flat
and open cosmological and dark matter models. The models are globally
normalized to fit cluster X-ray data, and intracluster gas is assumed to have
evolved in a manner consistent with current observations. We predict values of
y lower than the COBE/FIRAS upper limit. The corresponding values of the
overall optical thickness to Compton scattering are < 10^{-4} for relevant
parameter values. Of more practical importance are number counts of clusters
across which a net flux (with respect to the CMB) higher than some limiting
value can be detected. Such number counts are specifically predicted for the
COBRAS/SAMBA and BOOMERANG missions.
