Allocation of course seats to students is a challenging task for registrars' offices in universities. Since demand exceeds supply for many courses, course allocation needs to be done equitably and efficiently. Many schools use bidding systems where student bids are used both to infer preferences over courses and to determine student priorities for courses. However, this dual role of bids can result in course allocations not being market outcomes and unnecessary efficiency loss, which can potentially be avoided with the use of an appropriate market mechanism. We report a field experiment done at the University of Michigan Business School in Spring 2004 comparing its typical course bidding mechanism with the alternate Gale-Shapley Pareto-dominant market mechanism. Our results suggest that using the latter could vastly improve efficiency of course allocation systems while facilitating market outcomes.