Could a helium balloon float all the way up into space?

A helium-filled balloon can float very high up into the atmosphere, however, it cannot float up into outer space. The air in Earth's atmosphere gets thinner the higher up you go. The balloon can only rise up until the atmosphere surrounding it has the same weight as the helium in the balloon. This happens at about a height of 20 miles (32 kilometers) above Earth's surface. So, this is as far as a helium balloon can rise. Outer space starts somewhere around 600 miles (960 kilometers) above Earth's surface.