No, because 'light' is not a thing.
1. What we experience as light/shadow/color is an observer dependent sensation.
2. This subjective (observer dependent) sensation is mediated by wave-motion propagating through a medium of sub-atomic particles (commonly known as 'aether').
3. Therefore 'light' cannot be weighed because light is only a sensation much like sound, hot, cold, etc. are sensations. Perhaps you could find a way to 'weigh' the aether particles, but definitely not light.