It has recently been proposed that a mesoscopic ring of normal metal or semiconductor should exhibit equilibrium persistent currents of charge and spin, when embedded in an inhomogeneous magnetic field. The origin of these phenomena lies in the coupling between spin and orbital motion due to the Zeeman interaction, and the resulting geometric phase acquired during orbital motion. We present expressions for the ground state charge and spin currents for systems of many spin-1/2 fermions moving in a one-dimensional ring, and give numerical estimates of the magnitudes of the currents, and also of the electric field which results from the spin current.