A treaty supersedes all of the laws of the land, which is why it requires Congress to sign off on it, instead of a President's signature.
False. The Constitution is vague on the issue, saying that both treaties and the Constitution are "the law of the land". However, it's pretty clear that the Constitution only intended for treaties to deal with issues external to the United States (such as foreign trade).
In its only (to my knowledge) ruling which dealt with this issue, in regard to internal U.S. affairs, Reid v. Covert, SCOTUS ruled that the U.S. Constitution supersedes international treaties ratified by the U.S. Senate.
The actual "supremacy clause" in the Constitution reads thus:
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;
The phrases in pursuance thereof and under the authority of the United States clearly indicate that treaties may NOT supersede the Constitution. Because any such treaty, to be lawful, must be in pursuance of implementing the Constitution, and treaties must be made under the authority of the United States government, which is bound to uphold the Constitution.
So, NO. Claiming that a treaty supersedes internal U.S. laws or the Constitution is just false. It stems from a gross min-reading of the Supremacy Clause, and ignores past Supreme Court ruling on this issue.
The exception is when a State deals directly with foreign interests (such as international trade). In those cases, treaties do indeed supersede State laws on the matter, because the Federal government has all authority to deal with foreign governments and foreign trade. But that's a very narrow area.