I have to agree. Americans can't have dual loyalties therefore, they can't have dual citizenships.

Oath of Citizenship

“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 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.”

You are not considered a US citizen and until you take this oath. In addition to requiring your loyalty, this oath also requires you to possibly bear arms for the US and to give up any hereditary titles.