Adverse selection in procurement arises when low-cost bidders are also low-quality suppliers. We propose a mechanism called LoLA (lowball lottery auction) which, under some conditions, maximizes any combination of buyer's and social surplus, subject to incentive compatibility, in the presence of adverse selection. The LoLA features a floor price, and a reserve price. The LoLA has a dominant strategy equilibrium that, under mild conditions, is unique. In a counterfactual analysis of Italian government auctions, we compute the gain that the government could have made, had it used the optimal procurement mechanism (a LoLA), relative to a first-price auction (the adopted format).