Why does a helium balloon fly?

A balloon filled with helium rises up into the sky, while a balloon filled with air falls to the ground. What is the reason for these different kinds of reactions?

The reason for the ascension of the helium balloon is the buoyancy. Static buoyancy is a force which counteracts the weight force of the balloon. The buoyancy equals the weight force of the air which is displaced by the body. Every balloon has a certain volume. Therefore it displaces the same amount of air. (Archimedean Principle)

An example:

Let's say the balloon has a volume of 5 litres. Consequently it displaces 5 litres of air. Those approximately have a mass of 6.5 gramms, which is 0.0065 kg. Thus there is a weight force of
(F = m * g) 0,064 Newton.

How great is the lifting force acting on the balloon?