Existing solutions for exchange flow through straits rely upon the decomposition of the flow into a finite number of layers which have constant density. In this paper we provide a solution to inviscid steady exchange flow between continuously stratified reservoirs, where it is assumed that the flow in each direction is independently self-similar. The solution requires knowledge only of the two reservoir stratifications and an imposed net barotropic throughflow. The solution includes regions of stagnant fluid which separate two counter-flowing, stably stratified layers, with the provision that the two active layers may touch at no more than one point. Comparison of the theoretical solution with numerical simulations indicates that the assumption of self-similarity is reasonable, and that the disparity between the theoretical and simulated flows can be attributed to the inclusion of diffusion and viscosity in the numerical model.