Speaking of the Constitution, the paragraph just after that which you've cited from Article VI is interesting:
"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."
My interpretation, which seems to be supported by [1]: no treaty may expand the powers of government beyond those enumerated by the Constitution. If it is shown to (which seems likely given al the secrecy), it becomes unenforceable and we learn who to recall from office.
"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."
My interpretation, which seems to be supported by [1]: no treaty may expand the powers of government beyond those enumerated by the Constitution. If it is shown to (which seems likely given al the secrecy), it becomes unenforceable and we learn who to recall from office.
[1] http://caselaw.lp.findlaw.com/scripts/getcase.pl?navby=case&... (I found paragraph 2 of the opinion particularly salient to the topic), found from http://en.wikipedia.org/wiki/Article_Six_of_the_United_State...