IS a euro held in an Irish bank in Dublin, or in a Portuguese bank in Lisbon, as sound and secure as a euro in a German bank in Berlin? That apparently simple question holds the key to understanding why the euro zone may splinter and bring a new financial crisis.

In Ireland, there has been a “silent bank run” on financial institutions for much of the last year. In February, for instance, Irish private sector deposits dropped at an annual rate of 9.8 percent. That’s largely because some depositors doubt the commitment of the Irish government to the euro. They fear that they will wake up one morning to frozen bank accounts, followed by the conversion of their euro deposits into a lesser-valued new Irish currency. Pre-emptively, the depositors send their money outside Ireland, where it still represents safe euros or perhaps sterling, accessible by bank transfers and A.T.M. cards.

This flight of capital reflects a centuries-old economic principle known as Gresham’s Law, sometimes expressed casually as “bad money drives out good money.” In this context, if two assets — euros inside and outside Ireland — are not equal in value in the eyes of the marketplace, sooner or later the legally fixed price parity will fall apart.

If enough depositors fear frozen accounts, the banks will be emptied out, and they also will require additional government bailouts, on top of the bailouts for the bad real estate loans. The banks come to resemble empty shells, conduits for public aid but shrinking and unprofitable as businesses — and, to a large extent, that is already the case in Ireland. Portugal is moving in this same direction, toward being a land inhabited by zombie banks.