We consider the effect of an oscillating potential on the single-particle spectrum and the time-averaged persistent current of a one-dimensional phase-coherent mesoscopic ring with a magnetic flux. We show that in a ring with an even number of spinless electrons the oscillating potential has a strong effect on the persistent current when the excited side bands are close to the eigen levels of a pure ring. Resonant enhancement of side bands of the Floquet state generates a sign change of the persistent current.