  We study the flux-driven superconductor-metal transition in ultrasmall
cylinders observed experimentally by Liu {\em et.al.}(Science 294, 2332
(2001)). Where $T_c\to 0$, there is a quantum critical point, and a large
fluctuation conductivity is observed in the proximate metallic phase over a
wide range of $T$ and flux . However, we find that the predicted (Gaussian)
fluctuation conductivity in the neighborhood of the quantum critical point is 4
orders of magnitude smaller than observed experimentally. We argue that the
breakdown of Anderson's theorem at any non-integer flux leads to a broad
fluctuation region reflecting the existence of ``rare regions'' with local
superconducting order. We calculate the leading order correction to the
conductivity within a simple model of statistically induced Josephson coupled
local ordered regions that rationalizes the existing data.
