For a Coupled Map Lattice with a specific strong coupling emulating
Stavskaya's probabilistic cellular automata, we prove the existence of a phase
transition using a Peierls argument, and exponential convergence to the
invariant measures for a wide class of initial states using a technique of
decoupling originally developed for weak coupling. This implies the exponential
decay, in space and in time, of the correlation functions of the invariant
measures.