The equilibrium structure of a hard-sphere fluid confined in a small spherical cavity is investigated. In such systems the statistical mechanical ensembles are no longer equivalent and we consider both open (grand canonical) and closed (canonical) cavities in order to analyze the effects of size and packing constraints on the density profile of the confined fluid. For systems in the grand canonical ensemble the profiles are obtained from grand canonical ensemble Monte Carlo simulations and from density functional theory. The profiles of the closed (canonical) systems are obtained by means of canonical ensemble Monte Carlo simulations. A scheme is proposed which expands the canonical ensemble density profiles in terms of grand canonical averages; this is formally a series in powers of the inverse average number of particles. By comparing canonical ensemble Monte Carlo data with the results of the expansion applied to grand canonical ensemble Monte Carlo data and to the results of density functional theory the series expansion is shown to converge very quickly in most situations, even when the cavity contains only a few particles. However, as a consequence of packing constraints, in certain situations the density profile develops a pronounced peak in the center of the cavity. Then significant differences arise between the canonical and grand canonical profiles and the convergence of the series is much slower in the central zone where the peak develops. Describing accurately the various terms in the expansion and, hence, the detailed shapes of the profiles provides a searching test of density functional approximations. We find that recent modifications of Rosenfeld＇s fundamental measure theory, which are designed to describe situations of low effective dimensionality, perform better than his original theory and yield accurate results for all cases except those near maximum packing.