This paper shows that one of the de ning features of Walrasian equilibrium|law of one price|characterizes equilibrium in a non-Walrasian environment of (1) random trade matching without double coincidence of wants, and (2) strategic, price-setting conduct. Money is modeled as perfectly divisible and there is no constraint on agents' money inventories. In such an environment with discounting, the endogenous het- erogeneity of money balances among agents implies di erences in marginal valuation of money between distinct pairs of traders, which raises the question whether decen- tralized trade would typically involve price dispersion. We investigate the limiting case in which agents are patient, in the sense that they have overtaking-criterion preferences over random expected-utility streams. We show that in this case the \law of one price" holds exactly. That is, in a stationary Markov monetary equilibrium, all transactions endogenously must occur at a single price despite the decentralized organization of exchange. The result is in the same spirit as the work of Gale (1986a, b) on bargaining and competition, although the model di ers from Gale's in some signi cant respects.