The problem you have is knowing the pressure the helium is under, and so knowing how much helium is within the balloon. With any gas, if you know how much gas there is, it is a very simple calculation to know the mass of that gas - the relative masses of similar volumes of two gasses is always in proportion to their molecular weight.
Ofcourse, this is talking about mass rather than weight, and measured weight depends on the environment you are weighing the thing in. Weigh a helium balloon in air, and (assuming it has sufficient helium within the balloon, and not at too high a pressure or too low a temperature) it will have negative weight in air, but weight that same balloon in a hydrogen atmosphere and it will have positive weight.
If you weight the balloon in a helium atmosphere, and the pressure of the helium without is the same as the pressure of the helium within (which may not be easy to achieve, since the balloon itself, if it is a simply rubber type balloon, is likely to be put the helium under additional pressure no matter what the external pressure) then the only weight of the helium filled balloon would be that of the rubber balloon, since the helium within would balance the density of gas of the helium without.