  We consider the six-vertex model with the rational weights on an $s\times N$
square lattice, $s\leq N$, with partial domain wall boundary conditions. We
study the one-point function at the boundary where the free boundary conditions
are imposed. For a finite lattice, it can be computed by the quantum inverse
scattering method in terms of determinants. In the large $N$ limit, the result
boils down to an explicit terminating series in the parameter of the weights.
Using the saddle-point method for an equivalent integral representation, we
show that as $s$ next tends to infinity, the one-point function demonstrates a
step-wise behavior; at the vicinity of the step it scales as the error
function. We also show that the asymptotic expansion of the one-point function
can be computed from a second-order ordinary differential equation.
