  We consider the mesoscopic normal persistent current (PC) in a very
low-temperature superconductor with a bare transition temperature $T_c^0$ much
smaller than the Thouless energy $E_c$. We show that in a rather broad range of
pair-breaking strength, $T_c^0 \lesssim \hbar/\tau_s \lesssim E_c$, the
transition temperature is renormalized to zero, but the PC is hardly affected.
This may provide an explanation for the magnitude of the average PC's in the
noble metals, as well as a way to determine their $T_c^0$'s.
