Markovian imprecise jump processes provide a way to express model uncertainty about Markovian jump processes. The dynamics are not governed by a unique rate matrix, but are instead partially specified by a set of such matrices. Since the dynamics are partially specified, the resulting expected time averages are no longer uniquely determined either, and one then resorts to tight lower and upper bounds on them. In this paper, we are interested in the existence of an asymptotic limit of these upper and lower bounds, as the time horizon becomes infinite. When those limits exist and are furthermore independent of the choice of the process’s initial state, we say that the process is weakly ergodic. Our main contribution is a necessary and sufficient condition for a Markovian imprecise jump process to be weakly ergodic, expressed in terms of simple graph-theoretic conditions on its set of rate matrices.