  Consider a topologically transitive countable Markov shift $\Sigma$ and a
summable Markov potential $\phi$ with finite Gurevich pressure and
$\mathrm{Var}_1(\phi) < \infty$. We prove existence of the limit $\lim_{t \to
\infty} \mu_t$ in the weak$^\star$ topology, where $\mu_t$ is the unique
equilibrium state associated to the potential $t\phi$. Besides that, we present
examples where the limit at zero temperature exists for potentials satisfying
more general conditions.
