The Gilbert-Elliott channel, a varying binary symmetric channel, with crossover probabilities determined by a binary-state Markov process, is treated. In general, such a channel has a memory that depends on the transition probabilities between the states. A method of calculating the capacity of this channel is introduced and applied to several examples, and the question of coding is addressed. In the conventional usage of varying channels, a code suitable for memoryless channels is used in conjunction with an interleaver, with the decoder considering the deinterleaved symbol stream as the output of a derived memoryless channel. The transmission rate is limited by the capacity of this memoryless channel, which is often considerably less than the capacity of the original channel. A decision-feedback decoding algorithm that completely recovers this capacity loss is introduced. It is shown that the performance of a system incorporating such an algorithm is determined by an equivalent genie-aided channel, the capacity of which equals that of the original channel. The calculated random coding exponent of the genie-aided channel indicates a considerable increase in the cutoff rate over that of the conventionally derived memoryless channel.>