We propose a plasma experiment to be used to investigate fundamental properties of astrophysical dynamos. The highly conducting, fast-flowing plasma will allow experimenters to explore systems with magnetic Reynolds numbers an order of magnitude larger than those accessible with liquid-metal experiments. The plasma is confined using a ring-cusp strategy and subject to a toroidal differentially rotating outer boundary condition. As proof of principle, we present magnetohydrodynamic simulations of the proposed experiment. When a von K\'arm\'an-type boundary condition is specified, and the magnetic Reynolds number is large enough, dynamo action is observed. At different values of the magnetic Prandtl and Reynolds numbers the simulations demonstrate either laminar or turbulent dynamo action.