There’s not really any definite border. FAA controlled airspace generally ends at flight level 600 (around 60,000 feet, although flight levels are defined by air pressure not distance from the ground, so the actual altitude can vary). Above that will be uncontrolled airspace, though that doesn’t necessarily mean the US won’t claim sovereignty there.

The Outer Space treaty, generally considered the first step in establishing space law, stipulates that “outer space” is for the benefit of all mankind and not subject to sovereignty claims by any country. However there isn’t a legal definition of where outer space begins. The Karman line (100km) is a common practical definition. However the US has flown spacecraft (notably the shuttle) below 80km above Canada, without asking for permission first.

Practically speaking, there are as yet not enough craft flying at these kinds of altitudes for real legislation to be necessary. The spacefaring countries mostly work it out between them on a case by case basis.