We have established the solid-fluid coexistence region for a system of polydisperse hard spheres with near Gaussian diameter distributions, as a function of polydispersity. Our approach employs Monte Carlo simulation in the isobaric semigrand ensemble with a Gaussian activity distribution. Gibbs-Duhem integration is used to trace the coexistence pressure as a function of the variance of the imposed activity distribution. Significantly, we observe a "terminal" polydispersity above which there can be no fluid-solid coexistence. The terminus arises quite naturally as the Gibbs-Duhem integration path leads the pressure to infinity. This pressure divergence is an artifact of the method used to evaluate the freezing transition, because the sphere diameters vanish in this limit. A simple rescaling of the pressure with the average diameter brings the terminal pressure to a finite value. Nevertheless, the existence of this terminus only at infinite pressure precludes the construction of a continuous path from the solid to the fluid. At the terminus the polydispersity is 5.7% for the solid and 11.8% for the fluid while the volume fractions are 0.588 and 0.547 for the solid and fluid, respectively. Substantial fractionation observed at high values of the polydispersity (>5%) implies that the "constrained eutectic" assumption made in previous theoretical studies is not generally valid. Our results for the terminal polydispersity are consistent with experiments performed on polydisperse colloidal suspensions.