Joint source-channel coding for stationary memoryless and Gauss- Markov sources and binary Markov channels is considered. The channel is an additive-noise channel where the noise process is an M-th order Markov chain. Two joint source-channel coding schemes are considered. The first is a channel-optimized vector quantizer - optimized for both source and channel. The second scheme consists of a scalar quantizer and a maximum a posteriori detector. In this scheme, it is assumed that the scalar quantizer output has residual redundancy that can be exploited by the maximum a posteriori detector to combat the correlated channel noise. These two schemes are then compared against two schemes which use channel interleaving. Numerical results show that the proposed schemes outperform the interleaving schemes. For very noisy channels with high noise correlation, gains of 4 to 5 dB in signal-to-noise ratio are possible.