I Am Not A Lawyer (oh, so very not) but I have always read the various warnings to be found on the 'Web about how "treaties trump the U. S. Constitution" with no little skepticism. Didn't have a reason, the idea just didn't sit right; it is surely not what the Framers intended. Turns out there's some basis for that grain of salt: Reid v. Covert.
Back in 1957, the Supreme Court averred: "[T]his Court has regularly and uniformly recognized the supremacy of the Constitution over a treaty...."
Mind you, such decisions are not written in stone. It is, however, one more impediment to the kind of tricky end-runs around the Constitution and especially the Bill of Rights that folks worry over.
6 months ago