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The interpretation of galaxy number counts in terms of cosmological models isfraught with difficulty due to uncertainties in the overall galaxy population(mix of morphological types, luminosity functions etc.) and in the observations(loss of low surface brightness images, image blending etc.). Many of these canbe overcome if we use deep high resolution imaging of a single class of highsurface brightness galaxies, whose evolution is thought to be fairly wellunderstood. This is now possible by selecting elliptical and S0 galaxies usingHubble Space Telescope images from the Medium Deep Survey and other ultradeepWFPC2 images. In the present paper, we examine whether such data can be used todiscriminate between open and closed universes, or between conventionalcosmological models and those dominated by a cosmological constant. We find,based on the currently available data, that unless elliptical galaxies undergovery strong merging since $z \sim 1$ (and/or very large errors exist in themorphological classifications), then flat models dominated by a cosmologicalconstant are ruled out. However, both an Einstein-de Sitter ($\Omega_{0}=1$)model with standard passive stellar evolution and an open ($\Omega_{0}=0.05$)model with no net evolution ({\it i.e.} cancelling stellar and dynamicalevolution) predict virtually identical elliptical and S0 galaxy counts.  Based on these findings and the recent reportings of $H_{o} \simeq 80$ km/sMpc/s, we find that the maximum acceptable age of the universe is 13.3 Gyrs anda value of $\leq 9$ Gyrs favored. A flat---$\Lambda \neq 0$---universe istherefore {\it not} a viable solution to the $H_{o}$/globular cluster ageproblem.

Hubble Space Telescope Counts of Elliptical Galaxies: Constraints on Cosmological Models?