Actually google provides a much better definition...
notwithstanding /nätwiTHstandiNG/
Preposition
In spite of.
So if we read the applicable term from the constitution,
"and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding"
This is what it would say written in modern English...
"The Constitution, the Laws of the United States, and our International Treaty Obligations are the supreme law of the land. Judges are required to honor our treaty obligations; in spite of anything the Constitution or laws of any state may say to the contrary."