that's what i understand too Prudie. To hold dual citizenship, both nations have to agree and the US doesn't. When you take up US nationality you have to swear as follows.
"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."
The current exact text of the Oath of Citizenship is established only in the form of an administrative regulation promulgated by the executive branch. However, under the Administrative Procedure Act, CIS could theoretically change the text of the oath at any time, so long as the new text reasonably meets the "five principles" mandated by the Immigration and Nationality Act of 1953. These principles are:
allegiance to the United States Constitution,
renunciation of allegiance to any foreign country to which the immigrant has had previous allegiances
defense of the Constitution against enemies "foreign and domestic"
promise to serve in the United States Armed Forces when required by law (either combat or non-combat)
promise to perform civilian duties of "national importance" when required by law