  We consider the mean-field Zero-Range process in the regime where the
potential function $r$ is increasing to infinity at sublinear speed, and the
density of particles is bounded. We determine the mixing time of the system,
and establish cutoff. We also prove that the Poincar\'e constant is bounded
away from zero and infinity. This mean-field estimate extends to arbitrary
geometries via a comparison argument. Our proof uses the path-coupling method
of Bubley and Dyer and stochastic calculus.
