In uniform auctions, buyers choose demand schedules as strategies and pay the same "market clearing" price for units awarded. Despite the widespread use of these auctions, the extant theory shows that they are susceptible to arbitrarily large underpricing. We make a realistic modification to the theory by letting prices, quantities, and bids be discrete. We show that underpricing can be made arbitrarily small by choosing a sufficiently small price tick size and a sufficiently large quantity multiple. We also show how one might improve revenues by modifying the allocation rule. A trivial change in the design can have a dramatic impact on prices. Our conclusions are robust to bidders being capacity constrained. Finally, we examine supply uncertainty robust equilibria.