We study the endogenous choice to accept fiat objects as media of exchange and their implications for nominal exchange rate determination. We consider a two-country environment with two currencies which can be used to settle any transactions. However, currencies can be counterfeited at a fixed cost and the decision to counterfeit is private information. This induces equilibrium liquidity constraints on the currencies in circulation. We show that the threat of counterfeiting can pin down the nominal exchange rate even when the currencies are perfect substitutes, thus breaking the Kareken-Wallace indeterminacy result. When the two currencies are not perfect substitutes, an international currency can exist whereby one country has two currencies circulating while the other country uses only one. We also find that with appropriate fiscal policies we can enlarge the set of monetary equilibria with determinate nominal exchange rates. Finally, we show that the threat of counterfeiting can also help determine nominal exchange rates in a variety of different trading environments.