If it's pretty clear that a treaty would effectively repeal large parts of the constitution, you simply declare the treaty to be an executive agreement rather than a treaty, sign it, and begin enforcing the details. ACTA, the Anti-Counterfiting Trade Agreement, would grant sweeping powers to overseas entities to shut down internet sites without any legal oversight.
Under the provisions of ACTA, copyright holders will be granted sweeping direct powers to demand ISPs remove material from the Internet on a whim. Whereas ISPs normally are only forced to remove content after a court order, all legal oversight will be abolished, a precedent that will apply globally, rendering the treaty worse in its potential scope for abuse than SOPA or PIPA.Imagine elevating Righthaven to the status of global internet censor, with no accountability and you're getting close.
I find the potential precedent here to be rather unsettling. Imagine an "executive agreement" to forbid criticizing Islam, or limiting weapons possession. We doan' need no steenkin' Senate vote, now do we. It would probably be a good question for Eugine Volokh to ask what remedies would exist if the president declared this treaty to be an executive agreement and began enforcing it. Meantime, write your Senator and ask him or her if they plan on doing anything or if they're happy being highly paid and irrelevant.
I expect we'll be seeing a lot more of this sort of "rule by decree" effort in the last year of the Obama reign as his party increasingly sees him as a very lame duck who is likely to take people too close to him down with him.