The aim of this paper is to investigate the market efficiency on the foreign exchange market since the introduction of the Euro by applying the cointegration analysis to exchange rates. The introduction of the Euro has changed the structure of the global foreign exchange market to the extent that the second most important currency in the world with the highest credibility in the foreign exchange market, namely the Deutsche Mark, has been assimilated into the Euro. In order to evaluate if the introduction of a new currency has resulted in inefficient markets, a bivariate cointegration analysis should be applied to the seven most important exchange rates. The empirical analysis predominantly draws on the Johansen (1988, 1991) approach and the Gregory-Hansen (1996) approach whereas the latter takes endogenous structural breaks into account. We show that the foreign exchange market is broadly consistent with the market efficiency hypothesis. A very important result is that we can find a longrun relationship between the exchange rate pairs EUR/USD and GBP/USD whereas the no-arbitrage condition is satisfied. Since the EUR/USD exchange rate is weakly exogenous the GBP/USD exchange rate takes the burden of adjustment to the long-run equilibrium.