We study the behavior of configurations in the symmetric six-vertex model with $a,b,c$ weights in the $n\times n$ square with Domain Wall Boundary Conditions as $n\to\infty$. We prove that when $Δ=\frac{a^2+b^2-c^2}{2ab}<1$, configurations near the boundary have fluctuations of order $n^{1/2}$ and are asymptotically described by the GUE-corners process of the random matrix theory. On the other hand, when $Δ>1$, the fluctuations are of finite order and configurations are asymptotically described by the stochastic six-vertex model in a quadrant. In the special case $c=0$ (which implies $Δ>1$), the limit is expressed as the $q$-exchangeable random permutation of infinitely many letters, distributed according to the infinite Mallows measure.