I rather suspect the presence of a weighing scale complicates things a bit, so let's leave it out for a minute and let the balloon rest against the ceiling. What is the upward force on the ceiling? Per Archimedes' principle, it's the weight of the air displaced by the balloon less the weight of the balloon. At room temperature and pressure, a 1l balloon filled with helium would have a mass of about 0.17g, meaning a weight of about 1.7mN. The same volume of air has a mass of about 1.2g, meaning a weight of about 12mN. The upward force on the ceiling is therefore 12mN-1.7mN, or a little over 10mN, about the same as the weight of a 1g weight.
What would a balance show?
I think that a normal spring balance probably wouldn't work upside down, since it is likely to have mechanical stops to prevent the springs getting damaged by careless handling. I'd be a little surprised if the adjustment range were enough to let you zero it upside down, but it's not impossible. If it did, it would register a 10mN force, or probably 1g as they are usually calibrated to show mass assuming Earth-normal gravity.
You could certainly design a balance that would work upside down. Possibly the simplest would be something like one of the luggage-weighing devices I've seen around, which are basically a hand grip with a hook you can hang your suitcase from (although it would need to be a lot more precise than one of those to notice a 10mN force). Hang a weight from it, then zero it, then let the balloon push upwards on the weight. The result ought to be -1g. Alternatively, a classic scales-of-justice pan-balance with equal weights on each pan would do; again, let the balloon push up on one pan and see how much weight you have to remove (again, 1g) from the other pan to regain equilibrium.