Activated Random Walk (ARW) is an interacting particle system on the d-dimensional lattice Z^d. On a finite subset V of Z^d it defines a Markov chain on {0,1}^V. We prove that when V is a Euclidean ball intersected with Z^d, the mixing time of the ARW Markov chain is at most $1+o(1)$ times the volume of the ball. The proof uses an exact sampling algorithm for the stationary state, a coupling with...