Although a half-helium balloon floats better than a completely air-filled balloon, it will also deflate faster, based on principles governed by Graham's law, which holds that the rate at which gas escapes from a container is directly related to the molecular weight of the gas. Because helium is so much lighter than nitrogen and oxygen, the primary components of the atmosphere, it will escape at a faster rate. If you intend to keep your half-helium balloon airborne for a substantial length of time, you'll have to periodically replace the lost helium.
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It is possible to design a balloon that doesn't pop. If such a balloon were allowed to rise, it would eventually reach "the top of the atmosphere" and go no higher. Imagine that you release a balloon filled with air in the deep end of a swimming pool. The balloon will rise until it is floating on top of the water and go no higher. A balloon filled with helium would eventually get to the top of the atmosphere in that same way. It would depend on the weight of the balloon, but would be between 20 and 30 miles up.