"LOSS OF NATIONALITY: You may lose your United States nationality by being naturalized in, or by taking an oath or by making a declaration of allegiance to, a foreign state; or by serving in the armed forces or by accepting employment under the government of a foreign state. For detailed information, consult the nearest American consular office."
The next paragraph, while probably not pertinent in this case, looks like it would allow for combat duty in another nation's military if the person claims dual nationality. Another bag of worms.
"WARNING TO DUAL NATIONALS: A person is considered a dual national when he owes allegiance to more than one country at the same time. A claim to allegiance may be based on facts of birth, marriage, parentage, or naturalization. A dual national may, while in the jurisdiction of the other country which considers him its national, be subject to all of its laws, including military service. If difficulties occur, consult the nearest American consular office."