Treaties and the Constitution

In the Lord Monckton video posted below he states that if the US entered into the UN Climate Change treaty (which I doubt, even the overwhelming Democratic, Senate would ever ratify) that it would supercede the Constitution. This analysis argues that treaties do not, in fact, override the Constitution, which, as quotes provided at the end by Jefferson and Hamilton concur, would render the Constitution useless if it could be overridden by a treaty agreed to by 2/3rd of the Senate, a far less onerous task than amending the Constitution requires. That said, we have pointed out in this blog, the numerous occasions of clear violations of the Constitution that various Supreme Courts have turned a blind eye to or actively encouraged if it suited a desired political purpose. So if such a treaty was signed and directly conflicted with provisions of the Constitution I'm not sure we could depend on the SCOTUS to effectively nullify it.