How do you measure the volume of balloon by using a gram scale?

Hmmm. Tough problem. I don't think you can do it simply with a balance.

First off, I think you would need a balance that is much more accurate than a simple gram scale. You'd need an analytical balance (accurate to 0.0001 grams).

You would also need to know the ambient temperature and pressure and the tare weight of the balloon (weight of the uninflated balloon).

The weight in air would have to be corrected for the buoyancy of the air, but for that you would need the volume and that's what you're trying to find. Conundrum.

The pressure in the balloon would be slightly above the ambient pressure and that's difficult to measure. You could assume that it's the same as the atmospheric pressure, but in order to use that, you would have to measure the weight of the balloon in the absence of air -- and that would cause the balloon to pop.

You could put the balloon in liquid nitrogen and condense all of the oxygen. Then, you could try weighing it in a vacuum -- but you'd have to have some equipment that's very sophisticated. If you did get the weight, you could calculate the mass of oxygen and using the ideal gas law, you could calculate the volume of gas.

Phew!

An easier way (to get a very accurate volume) would be to fill a very large graduated cylinder with water (graduated cylinder diameter larger than the diameter of the balloon). Pull the balloon under water down into the cylinder and measure the volume on the graduated cylinder. Subtract that from the volume of water in the cylinder before you put the balloon in there. The difference is the volume of the balloon. There would be a very slight correction for pressure changes.

The easiest way, of course, is simply to measure the balloon's circumference and calculate the volume (probably assuming a sphere).

Look through here:
http://www.wbrschools.net/techcds/webpage%20math%2056/data/are%20you%20full%20of%20hot%20air%20g6gle31.pdf
Unless baloon is a perfect circle, you could measure by filling with water, the same size, and measure water volume?
Then you could measure the volume of water with the Gram scale?
As I am adding, I notice Perito's mention above of water.
I feel you can get a more accurate measurement of volume, by measuring though weight of water as opposed to the weight of air?

Another thought, fill a container with water and submurge baloon to see the volume of water it displaced, measure?

If you know the density of air it would be m/d to get the volume
And yes you would need a better scale