We consider an assemble-to-order system with a high volume of prospective customers arriving per unit time. Our objective is to maximize expected infinite-horizon discounted profit by choosing product prices, component production capacities, and a dynamic policy for sequencing customer orders for assembly. We prove that a myopic discrete-review sequencing policy, which allocates scarce components among orders for different products to minimize instantaneous physical and financial holding costs, is asymptotically optimal. Furthermore, we prove that optimal prices and production capacity nearly balance the supply and demand for components (i.e., it is economically optimal to operate the system in heavy traffic), so system performance is characterized by a diffusion approximation. The diffusion approximation exhibits state-space collapse: Its dimension equals the number of components (rather than the number of components plus the number of products). These results complement the existing assemble-to-order literature, which focuses on managing component inventory and assumes FIFO sequencing of orders for assembly.