We analyze a bilateral trade model where the buyer chooses the distribution of her valuation for the good. The seller, after observing the buyer’s distribution but not the realized valuation, makes a take-it-or-leave-it offer. If distributions are costless, the price and the payoffs of both the buyer and the seller are shown to be 1/e in the unique equilibrium outcome. The buyer’s equilibrium distribution generates a unit-elastic demand, and trade is ex post efficient. These properties are shown to be preserved even when different distributions are differentially costly as long as the cost is monotone in the dispersion of the distribution.