We obtain a tight bound of O(L2logk) for the mixing time of the exclusion process in Zd/LZd with k≤½Ld 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(L2logd) improves on the earlier bound O(L2d) obtained by Quastel. Our proof is based on an auxiliary Markov chain we call the chameleon process, which may be of independent interest.