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Channel dispersion plays a fundamental role in assessing the backoff from capacity due to finite blocklength. This paper analyzes the channel dispersion for a simple channel with memory: the Gilbert-Elliott communication model in which the crossover probability of a binary symmetric channel evolves as a binary symmetric Markov chain, with and without side information at the receiver about the channel state. With side information, dispersion is equal to the average of the dispersions of the individual binary symmetric channels plus a term that depends on the Markov chain dynamics, which do not affect the channel capacity. Without side information, dispersion is equal to the spectral density at zero of a certain stationary process, whose mean is the capacity. In addition, the finite blocklength behavior is analyzed in the non-ergodic case, in which the chain remains in the initial state forever.