Buoyancy affects measured weight in which case the above answers are correct where the same type and density of gas is found within and outside of the balloon, however, (as whowhatwhy mentioned) the elasticity of the balloon will compress the internal gas slightly thereby giving the gas inside a slightly greater pressure/density, reduced buoyancy and therefore slightly increased weight when inflated.
Helium has a lower mass/density than an equal volume of air under similar pressure and therefore increases buoyancy and reduces the weight as measured in a standard atmosphere (even after slight compression).
When measured within a vacuum (where buoyancy is no longer a factor) the contents of the balloon will increase its measured weight in direct proportion to the mass of the added contents, however, with no external pressure a comparatively small quantity of gas will rupture the balloon.
whowhatwhy, guess I better wipe after that one!