Well, this is what I've found and this is what I understand to be true.
A description of the US naturalization oath is given in Section
337(a) of the INA [
8 USC § 1448(a)]. Of particular relevance to the dual citizenship issue is that, as part of the oath, a new citizen must pledge "
to renounce and abjure absolutely and entirely all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which the applicant was before a subject or citizen."
In practice, it is unclear what if any true legal significance this statement has any more. The US does not require a new citizen to take any formal steps to renounce his old citizenship before officials of the "old country"; and when the other country continues to claim a naturalized US citizen as one of its own, current
US policy recognizes that such a person may have to use a passport from the other country in order to visit there, and such an action does
not put the person's US citizenship in jeopardy.