  In the present article we obtain the large $N$ asymptotics of the partition
function $Z_N$ of the six-vertex model with domain wall boundary conditions on
the critical line between the disordered and antiferroelectric phases. Using
the weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\rightarrow\infty$,
$Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit
expression in $x$ and the $x$-dependency in $C$ is determined. This result
reproduces and improves the one given in the physics literature by Bogoliubov,
Kitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an
infinite order phase transition between the disordered and antiferroelectric
phases. Our proofs are based on the large $N$ asymptotics for the underlying
orthogonal polynomials which involve a non-analytical weight function, the
Deift-Zhou nonlinear steepest descent method to the corresponding
Riemann-Hilbert problem, and the Toda equation for the tau-function.
