  We consider a two-dimensional singularly perturbed transmission problem with
two different diffusion coefficients, in a domain with smooth (analytic)
boundary. The solution will contain boundary layers only in the part of the
domain where the diffusion coefficient is high and interface layers along the
interface. Utilizing existing and newly derived regularity results for the
exact solution, we design a robust $hp$ finite element method for its
approximation. Under the assumption of analytic input data, we show that the
method converges at an e{xponential} rate, provided the mesh and polynomial
degree distribution are chosen appropriately. Numerical results illustrating
our theoretical findings are also included.
