"We have created a superfluid atom circuit using a toroidal Bose-Einstein Condensate. Just as a current in a superconducting circuit will flow forever, if a current is created in our superfluid circuit, the flow will not decay as long as the current is below a critical value. A repulsive optical barrier across one side of the torus creates the tunable weak link in the condensate circuit and can be used to control the current around the loop. By rotating the weak link at low rotation rates, we have observed phase slips between well-defined persistent current states. This behavior is analogous to that of a weak link in a superconducting loop. A feature of our system is the ability to dynamically vary the weak link, which in turn varies the critical current, a feature that is difficult to implement in superconducting circuits. For higher rotation rates, we observe a transition to a regime where vortices penetrate the bulk of the condensate. These results demonstrate an important step toward realizing an atomic SQUID analog."