At present, the European Union (EU) is facing the biggest round of enlargement in its history. Many of the EU accession countries, which are less developed than the incumbents, will seek accession to the European Monetary Union (EMU) soon after the required two-year qualifying period. This is justified by the effort to avoid the danger of financial instability in the period prior to euro-introduction. However, by trying to avoid this danger, some of them may run into another danger, namely of a lack of real convergence. The paper investigates this danger.