  Square ice is a statistical mechanics model for two-dimensional ice, widely
believed to have a conformally invariant scaling limit. We associate a Peano
(space filling) curve to a square ice configuration, and more generally to a
so-called 6-vertex model configuration, and argue that its scaling limit is a
space-filling version of the random fractal curve SLE$_\kappa$,
Schramm--Loewner evolution with parameter $\kappa$, where $4<\kappa\leq
12+8\sqrt{2}$. For square ice, $\kappa=12$. At the "free-fermion point" of the
6-vertex model, $\kappa=8+4\sqrt{3}$. These unusual values lie outside the
classical interval $2\le \kappa\le 8$.
