We have produced persistent currents of ultracold fermionic atoms trapped in a toroidal geometry with lifetimes greater than 10 seconds in the strongly-interacting limit. These currents remain stable well into the BCS limit at sufficiently low temperature. We drive a circulating BCS superfluid into the normal phase and back by changing the interaction strength and find that the probability for quantized superflow to reappear is remarkably insensitive to the time spent in the normal phase and the minimum interaction strength. After ruling out the Kibble-Zurek mechanism for our experimental conditions, we argue that the reappearance of superflow is due to long-lived normal currents and the Hess-Fairbank effect.