This paper develops a discrete choice model of demand with optimal sequential consumer search. Consumers first choose a product to search; then, once they learn the utility they get from the searched product, they choose whether to buy it or to keep searching. We characterize the search problem as a standard discrete choice problem and propose a parametric search cost distribution that generates closed-form expressions for the probability of purchasing a product. We propose a method to estimate the model that supplements aggregate product data with individual-specific data which allows for the separate identification of search costs and preferences. We estimate the model using data from the automobile industry and find that search costs have non-trivial implications for elasticities and markups. We study the effects of exclusive dealing regulation and find that firms benefit at the expense of consumers, who face higher search costs and higher prices than would be the case if multi-brand dealerships were used.