The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)

We prove that the height function of the square-lattice six-vertex model with weights $a=b=1$ and $0<c\le 2$, in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel $-(2\pi {)}^{-1}\log |x-y|$, the squared multiplier is $1/\arcsin (c/2)$.