there is a deep tug-of-war in this. The EU position is that European Law is higher level than any law of member states, including their constitutions. States, generally, say that EU law takes its very legitimation from the souce of national law and, consequently, cannot overrule it.
I wouldn't worry about that, here in Europe we have a long tradition and history of solving our problems in a conciliatory and, above all else, peaceful way.