“There is not much for us in Romania,” she said recently, watching her husband sleep. “And now that we are in the European Union, we have the right to go to other countries. It is better there.”

Thousands of Romania’s Roma, also known as Gypsies, have come to a similar conclusion in recent years, heading for the relative wealth of Western Europe, and setting off a clash within the European Union over just how open its “open borders” are.