  We analyze the crossover from an hc/e-periodicity of the persistent current
in flux threaded clean metallic rings towards an hc/2e-flux periodicity of the
supercurrent upon entering the superconducting state. On the basis of a model
calculation for a one-dimensional ring we identify the underlying mechanism,
which balances the hc/e versus the hc/2e periodic components of the current
density. When the ring circumference exceeds the coherence length of the
superconductor, the flux dependence is strictly hc/2e periodic. Further, we
develop a multi-channel model which reduces the Bogoliubov - de Gennes
equations to a one-dimensional differential equation for the radial component
of the wave function. The discretization of this differential equation
introduces transverse channels, whose number scales with the thickness of the
ring. The periodicity crossover is analyzed close the critical temperature.
