  We obtain a tight bound of $O(L^2\log k)$ for the mixing time of the
exclusion process in $\mathbf{Z}^d/L\mathbf{Z}^d$ with $k\leq{1/2}L^d$
particles. Previously the best bound, based on the log Sobolev constant
determined by Yau, was not tight for small $k$. When dependence on the
dimension $d$ is considered, our bounds are an improvement for all $k$. We also
get bounds for the relaxation time that are lower order in $d$ than previous
estimates: our bound of $O(L^2\log d)$ improves on the earlier bound $O(L^2d)$
obtained by Quastel. Our proof is based on an auxiliary Markov chain we call
the chameleon process, which may be of independent interest.
