If you have read the article How Helium Balloons Work, then you know that a liter of air at sea level weighs about 1.25 grams. A liter is 1,000 cubic centimeters, or about 61 cubic inches -- the size of a 1-liter soda bottle. A liter of helium, on the other hand, weighs about 0.18 grams. If you weigh a 1-liter bottle filled with air and then weigh the same bottle filled with helium, it will weigh about 1.07 grams less. If the bottle itself weighed less than a gram, you couldn't weigh it at all -- it would float! You could turn the scale upside down and put it above the floating bottle to check its negative weight! Generally, a balloon has to be several liters in size before the 1-gram-per-liter weight difference of helium vs. air is enough to overcome the weight of the balloon itself and float.
If you could somehow fill a 1-liter bottle with a vacuum, it would float even better. A perfect vacuum weighs zero grams, so a liter of perfect vacuum weighs 0.18 grams less than a liter of helium. The problem, of course, is that building a lightweight container that can hold a vacuum is not nearly as easy as building a fabric envelope that can hold helium. The phrase "Nature abhors a vacuum" sums it up nicely. If you could figure out a way to do it, however, you would be set -- your vacuum balloon would float!
Note that you would not need to have a perfect vacuum. Any air that you take out of the envelope will lower the weight and cause lift. You might be able to build some sort of funky fabric envelope to hold a partial vacuum. For example, imagine taking a large sphere and filling it with pressurized air. Inside the large sphere you place a slightly smaller sphere and you attach the inner sphere to the outer sphere with many strings or with fabric baffles. If you were to create a vacuum in the inner sphere, the pressurized gas between the inner and outer sphere would keep the outer sphere's shape. The strings or fabric holding the inner sphere to the outer sphere would keep the inner sphere from collapsing. You could develop a partial vacuum inside the inner sphere and the whole structure might float... Who knows?
It's probably easier to just use helium.
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