The stability of the body-centered cubic (bcc) solid phase of classical hard spheres is of intrinsic interest and is also relevant to the development of perturbation theories for bcc solids of other model systems. Using canonical ensemble Monte Carlo, we simulated systems initialized in a perfect bcc lattice at various densities in the solid region. We observed that the systems rapidly evolved into one of four structures that then persisted for the duration of the simulation. Remarkably, one of these structures was identified as cI16, a cubic crystalline structure with 16 particles in the unit cell, which has recently been observed experimentally in lithium and sodium solids at high pressures. The other three structures do not exhibit crystalline order but are characterized by common patterns in the radial distribution function and bond-orientational order parameter distribution; we refer to them as bcc-di, with i ranging from 1 to 3. We found similar outcomes when employing any of the three single occupancy cell (SOC) restrictions commonly used in the literature. We also ran long constant-pressure simulations with box shape fluctuations initiated from bcc and cI16 initial configurations. At lower pressures, all the systems evolved to defective face-centered cubic (fcc) or hexagonal close-packed (hcp) structures. At higher pressures, most of the systems initiated as bcc evolved to cI16 with some evolving to defective fcc/hcp. High pressure systems initiated from cI16 remained in that structure. We computed the chemical potential of cI16 using the Einstein crystal reference method and found that it is higher than that of fcc by ∼0.5kT-2.5kT over the pressure range studied, with the difference increasing with pressure. We find that the undistorted bcc solid, even with constant-volume and SOC restrictions applied, is so mechanically unstable that it is unsuitable for consideration as a metastable phase or as a reference system for studying bcc phases of other systems. On the other hand, cI16 is a mechanically stable structure that can spontaneously emerge from a bcc starting point but it is thermodynamically metastable relative to fcc or hcp.