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Given a discrete stationary channel for which the map carrying each stationary, ergodic input into the input-output measure is continuous (with respect to weak convergence) at at least one input, it is shown that every stationary and ergodic source with sufficiently small entropy is block transmissible over the channel. If this weak continuity condition is satisfied at every stationary ergodic input, one obtains the class of weakly continuous channels for which the usual source/channel block coding theorem and converse hold with the usual notion of channel capacity. An example is given to show that the class of weakly continuous channels properly includes the class of -continuous channels. It is shown that every stationary channel is "almost" weakly continuous in the sense that every input-output measure for can be obtained by sending over an appropriate weakly continuous channel (depending on ). This indicates that weakly continuous channels may be the most general stationary channels for which one would need a coding theorem.