  We study the large time behavior of solutions to a linear transmission
problem in one space dimension. The problem at hand models a thermoelastic
material with second sound confined by a purely elastic one. We shall
characterize all equilibrium states of the considered system and prove that
every solution approaches one designated equilibrium state with an exponential
rate as time goes to infinity. Hereto, we apply methods from the theory of
strongly continuous semigroups. In particular, we obtain uniform resolvent
bounds for the underlying generator. This removes the largeness assumption of
elastic wave speeds imposed in [Y.P. Meng and Y.G. Wang, Anal. Appl. (Singap.)
13 (2015)] for having an exponential energy decay rate when the problem only
has the trivial equilibrium. In an appendix we provide a similar exponential
stability result for the case where heat conduction is modeled using Fourier's
law.
