There seems to be an idea floating around that treaties can trump the Constitution. Eugene Volokh even mentions it here. Now, of course, there's no telling how activist judges of the future may interpret the Constitution (which seems to be Mr. Volokh's fear), but from a direct reading of the text it's hard to see much cause for concern. Paragraph 2 of Article VI says:
This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.Thus, "laws of the United States" and "treaties" are put into the exact same category, as is appropriate. The former are proposed by Congress and signed by the President, and the latter are proposed by the President and ratified by the Senate -- there's no reason to think that either should have more authority than the other.
Further, even if one were to somehow conclude that treaties could be used to nullify portions of the Constitution, the Bill of Rights and subsequent amendments would be immune. Why? Because the authority of any treaty would rest here in Article VI, and when amendments are passed they override any pre-existing Constitutional obstacle to their enforcement. Just as the Thirteenth Amendment invalidated portions of Article IV, the Bill of Rights and subsequent amendments would have to be held to invalidate any interpretation or use of the treaty clause that opposed their enforcement.