  Consider a H\"older continuous potential $\phi$ defined on the full shift
$A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified
sofic subshift. It is well-known that there is a unique Gibbs measure
$\mu_\phi$ on $X$ associated to $\phi$. Besides, there is a natural nested
sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift
$X$. To this sequence we can associate a sequence of Gibbs measures
$(\mu_{\phi}^m)$. In this paper, we prove that these measures weakly converge
at exponential speed to $\mu_\phi$ (in the classical distance metrizing weak
topology). We also establish a strong mixing property (ensuring weak
Bernoullicity) of $\mu_\phi$. Finally, we prove that the measure-theoretic
entropy of $\mu_\phi^m$ converges to the one of $\mu_\phi$ exponentially fast.
We indicate how to extend our results to more general subshifts and potentials.
We stress that we use basic algebraic tools (contractive properties of iterated
matrices) and symbolic dynamics.
