We analyze the coupling of the dual-mixed finite element method with the boundary integral equation method. The result is a new mixed scheme for the exterior Stokes problem. The approach is based on the introduction of both the flux and the strain tensor of the velocity as further unknowns, which yields a two-fold saddle point problem as the resulting variational formulation. We show existence and uniqueness of the solution for the continuous and discrete formulations and provide the associated error analysis. In particular, the corresponding Galerkin scheme is defined by using piecewise constant functions and Raviart-Thomas spaces of lowest order. Most of our analysis makes use of an extension of the classical Babuška-Brezzi theory to a class of saddle-point problems. Also, we develop a-posteriori error estimates (of Bank-Weiser type) and propose a reliable adaptive algorithm to compute the finite elements solutions. Finally, several numerical results are given.