EricM,
You are completely and totally wrong. Actually the US does not allow you to receive American citizenship unless you renounce all other foreign citizenship.
HOWEVER, once you have US citizenship, however it was acquired, it is very difficult to get rid of and acquiring or being granted foreign citizenship does not, according to the State department, even suggest you wish to renounce US citizenship. Renouncing your citizenship requires a formal statement at a US embassy. Even then, if the IRS believes you are renouncing citizenship in order to avoid our punitive tax system, they will claim the authority to tax you as if you are a US citizen.
In the past it was suggested that serving in foreign military forces was evidence you intended to renounce US citizenship, but today this is not considered the case. Witness the many US/Israeli citizens who serve in the Israeli military.
I personally know many dual citizens, and they have all the legal rights and official documents that you or I have.