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