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In this paper, we consider ergodic causal classical-quantum channels (cq-channels) which additionally have a decaying input memory. In the first part, we develop some structural properties of ergodic cq-channels and provide equivalent conditions for ergodicity. In the second part, we prove the coding theorem with weak converse for causal ergodic cq-channels with decaying input memory. Our proof is based on the possibility to introduce a joint input-output state for the cq-channels and an application of the Shannon-McMillan theorem for ergodic quantum states. In the last part of the paper, it is shown how this result implies a coding theorem for the classical capacity of a class of causal ergodic quantum channels.