We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the Mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the Mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the Mirror Langevin dynamics including the Mirror Langevin algorithm have an asymptotic bias. For this algorithm, we also give upper bounds for the number of iterations taken to mix to a constrained distribution whose potential is relatively smooth, convex, and Lipschitz continuous with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the inclusion of the Metropolis-Hastings filter, we obtain an exponentially better dependence on the error tolerance for approximate constrained sampling. We also present numerical experiments that corroborate our theoretical findings.