This paper provides a rigorous proof of the connection between the optimal sequencing problem for a two-station, two-customer-class queueing network and the problem of control of a multidimensional diffusion process, obtained as a heavy traffic limit of the queueing problem. In particular, the diffusion problem, which is one of "singular control" of a Brownian motion, is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network. The results are proved by a viscosity solution analysis of the related Hamilton-Jacobi-Bellman equations.