The subject of this paper is a two-station mixed queueing network with two customer types: "Open" customers enter the network at station 1 and depart the system after receiving service. Meanwhile, a fixed number of "closed" customers circulate between stations 1 and 2 indefinitely. Such a mixed queueing network model can represent a single-stage production system that services both make-to-order and make-to-stock customers. We present fluid and diffusion limits for this network under the first-in-first-out service discipline. We find that the heavy traffic limit of the workload process at station 1 is a reflected Brownian motion (RBM) on a finite interval.This result is surprising in light of the behavior of the original mixed network model, in which the workload at station 1 need not be bounded.