Well, apparently I am wrong. Although in the Oath of Allegiance (see below) the new US citizen renounces his or her previous citizenship, this is not enforced by the US presently and more and more non-US countries are choosing to not dissolve the old citizenship.
This was not your question anyway, you were going from US to other, not other to US.
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Dual Citizenship, the Naturalized US Citizen, and the Oath of Allegiance
"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" from the Oath of Allegiance
The opening lines of the Oath of Allegiance are meant to give the United States exclusive sovereignty over the newly naturalized citizen. In other words, you are a citizen of one and only one country, the United States of America. The idea is that as soon as you take the Oath of Allegiance and become an American, you are giving up your citizenship of your native country.
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