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\ge1. 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.
