This paper studies the transaction cost savings of moving from a multi-currency exchange system to a single currency one. The analysis concentrates exclusively on the transaction and precautionary demand for money and abstracts from any other motives to hold currency. A continuous-time, stochastic Baumol- like model similar to that in Frenkel and Jovanovic (1980) is generalized to include several currencies and calibrated to fit European data. The analysis implies an upper bound for the savings associated with reductions of transaction costs derived from the European Monetary Union of approximately 0.6\% of the Community GDP. Additionally, the magnitudes of the brokerage fee and the volatility of transactions, whose estimation has traditionally been difficult to address empirically, are approximated for Europe.