We examine the last unexplored regime of the asymmetric six-vertex model: the low-temperature phase of the so-called ferroelectric model. The original publication of the exact solution by Sutherland, Yang, and Yang and various derivations and reviews published afterward do not contain many details about this regime. We study the exact solution for this model by numerical and analytical methods. In particular, we examine the behavior of the model in the vicinity of an unusual coexistence point that we call the "conical" point. This point corresponds to additional singularities in the free energy that were not discussed in the original solution. We show analytically that at this point many polarizations coexist, and that unusual scaling properties hold in its vicinity.