International Law as Law in the United States

"International law is part of our law." Justice Gray's much-quoted pronouncement in The Paquete Habana was neither new nor controversial when made in 1900, since he was merely restating what had been established principle for the fathers of American jurisprudence and for their British legal ancestors. And Gray's dictum remains unquestioned today. But, after more than two hundred years in our jurisprudence, the import of that principle is still uncertain and disputed. How did, and how does, international law become part of our law?

What does it mean that international law is a part of our law? What is the relation of that part of our law to other parts of our law?