In an assemble-to-order system, a wide variety of products are rapidly assembled from component inventories in response to customer orders. We assume that orders must be filled within a product-specific target lead time. In the event that some of the components required to fill an order are out of stock, these components must be expedited. The objective is to minimize the expected infinite-horizon discounted cost of primary component production and expediting. Our formulation captures financial holding costs but implicitly assumes that physical holding costs are negligible. The controls are (1) sequencing orders for assembly, (2) primary component production, and (3) component expediting. We prove that the multidimensional assemble-to-order control problem separates into single-item inventory control problems. In particular, under an optimal policy for assembly sequencing, the optimal production and expediting policy for each component is independent of all other components. Hence, the literature on single-item inventory management with expediting or lost sales is directly relevant to the control of assemble-to-order systems.