This is a topic that often leads to confusion. If you are a U.S. citizen and you wish to become a citizen of another country you are not required to renounce your U.S. citizenship unless the other country requires it.

However, if you are a foreign national and you wish to become a U.S. citizen you do in fact have to renounce your other citizenship. Here is the oath you must take to become a U.S. citizen:

"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty, of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely, without any mental reservation or purpose of evasion; so help me God."

http://www.uscis.gov/us-citizenship/...states-america

If you were born in the U.S. you take no oath.

As noted above, U.S. law does not recognize dual nationality - but doesn't prohibit it either. U.S. law is simply silent on the issue. So there is a double standard - if you were born a U.S. citizen and wish to add a second citizenship, unless the new country requires otherwise, you can be a citizen of both (and in some circumstances more than two). However, if you are a foreign national and you want to become a U.S. citizen you must renounce all other citizenships.

However, in the case of a U.K. national, the U.K. deems such renounciation to be made under duress and ineffective. Thus, from the U.S. perspective you renounced but from the U.K. perspective you did not do so effectively and you can maintain a U.K. passport. And since U.S. law is silent on the issue you can exist in that ambiguous state.