  We show that the height function of the six-vertex model, in the parameter
range $\mathbf a=\mathbf b=1$ and $\mathbf c\ge1$, is delocalized with
logarithmic variance when $\mathbf c\le 2$. This complements the earlier proven
localization for $\mathbf c>2$. Our proof relies on Russo--Seymour--Welsh type
arguments, and on the local behaviour of the free energy of the cylindrical
six-vertex model, as a function of the unbalance between the number of up and
down arrows.
