In Hotelling's two-stage model of spatial competition, two firms simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer), it is known that the price-setting subgames posses equilibria in pure strategies for only a limited set of location pairs, so that the two-stage game has no equilibrium in pure strategies. This paper shows that, for a subset of location pairs, all equilibria of the second stage of the game in which the players use mixed strategies are of a certain type. For a large finite number of location pairs, computational methods are used to find approximate equilibria of this type. The approximate second-stage equilibria are used to compute an approximate equilibrium of the whole game. In this equilibrium, the firms locate just inside the quartile of the market; they randomize over prices, giving most weight to a relatively high price.