The reading will go up by an amount equivalent to the weight of water displaced by the stone.
Archimedes principle states that a body immersed in a fluid experiences an upward force equal to the weight of fluid displaced (that is the weight of fluid of the same volume as the object). then states that for every force there is an equal and opposite reaction. That is, the same fluid displacement force must act down on the fluid and therefore on the container and therefore adds to the weight..
As an aside, the masslessness of the string is irrelevant, what is important is that it has negligible volume in the water.
