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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 that 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 famous Kareken-Wallace indeterminacy result.