The equilibrium vacancy concentration in solids can be computed from density-functional theory (DFT) if allowance is made for density profiles with less than one particle per lattice site. For the fundamental-measure theory (FMT), this approach predicts reasonably small vacancy concentrations in hard sphere crystals, in contrast to earlier DFTs. Using an asymptotic analysis of the FMT functional, it is shown that the number of vacancies depends exponentially on the distance to the close packing density, as expected from heuristic arguments. The prefactor of the exponential is calculated for three recently suggested variants of the theory, using density profiles obtained from a quasifree minimization. Extrapolation of the asymptotic behavior to the melting density yields good agreement with other estimates and computer simulation results.