  This paper is concerned with the controller-and-stopper stochastic
differential game under a regime switching model in an infinite horizon. The
state of the system consists of a number of diffusions \emph{coupled} by a
continuous-time finite-state Markov chain. There are two players, one called
the controller and the other called the stopper, involved in the game. The goal
is to find a saddle point for the two players up to the time that the stopper
\emph{terminates} the game. Based on the dynamic programming principle (DPP,
for short), the lower and upper value functions are shown to be the viscosity
supersolution and viscosity subsolution of the associated
Hamilton-Jacobi-Bellman (HJB, for short) equation, respectively. Further, in
view of the comparison principle for viscosity solutions, the lower and upper
value functions \emph{coincide}, which implies that the game admits a value.
All the proofs in this paper are strikingly different from those for the case
without regime switching.
