  The six-vertex model, or the square ice model, with domain wall boundary
conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and
Izergin. The solution is based on the Yang-Baxter equations and it represents
the free energy in terms of an $N\times N$ Hankel determinant. Paul Zinn-Justin
observed that the Izergin-Korepin formula can be re-expressed in terms of the
partition function of a random matrix model with a nonpolynomial interaction.
We use this observation to obtain the large $N$ asymptotics of the six-vertex
model with DWBC in the disordered phase. The solution is based on the
Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method.
As was noticed by Kuperberg, the problem of enumeration of alternating sign
matrices (the ASM problem) is a special case of the the six-vertex model. We
compare the obtained exact solution of the six-vertex model with known exact
results for the 1, 2, and 3 enumerations of ASMs, and also with the exact
solution on the so-called free fermion line. We prove the conjecture of
Zinn-Justin that the partition function of the six-vertex model with DWBC has
the asymptotics, $Z_N\sim CN^\kappa e^{N^2f}$ as $N\to\infty$, and we find the
exact value of the exponent $\kappa$.
