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I fully agree that treaties signed by the President and ratified by the Senate become US law, so long as they are constitutional. The clause of the Vienna treaty you cite is unconstitutional because it conflicts with how the constitution specifies treaties become US law. So, the president's signature alone may be enough for other countries to consider us a party to a treaty but the USSC would laugh you out of court if you tried to cite such a treaty to them as governing law, and rightfully so.