  In the present paper we obtain rates of convergence for the limit theorems of
the density vector in the Curie-Weiss-Potts model via Stein's Method of
exchangeable pairs. Our results include Kolmogorov bounds for multivariate
normal approximation in the whole domain $\beta\geq 0$ and $h\geq 0$, where
$\beta$ is the inverse temperature and $h$ an exterior field. In this model,
the critical line $\beta = \beta_c(h)$ is explicitly known and corresponds to a
first order transition. We include rates of convergence for non-Gaussian
approximations at the extremity of the critical line of the model.
