No, that would knock the sponsor’s neon signs off the sides, and that would never do because today’s blimps are nothing but flying billboards. Instead, they use a clever system of swapping helium and air back and forth.
The blimp is, as you’ve noted, essentially a big rubber bag full of helium. The contraption floats in the air because the whole thing, helium, rubber bag, gondola, engine, crew, and joyriding local politicians, together weigh less than an equal volume of air.
On a hot day with the sun beating down on it, there can be quite a pressure buildup, tending to make the helium expand. But they can’t just vent all that expensive helium out into the air. Moreover, what would they do when the bag cools and they need more helium to keep it, from looking like a flying prune?
There is a small, separate bag of air inside the big bag of helium, like an air balloon inside a helium balloon. They are arranged so that when the helium expands, it just pushes some cheap, old air out of the ship.
And when the helium contracts, they make up for the shrinkage by blowing more air into the inner bag. Or else they get the politicians to make speeches into it.