The six-vertex model, also known as the square ice model, with domain wall boundary condition (DWBC) has been introduced and solved for finite N by Korepin and Isergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an NxN Hankel determinant. Paul Zinn-Justin observed that the IserginKorepin 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.