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This is a continuation of the papers of Bleher and Fokin (Commun. Math. Phys., 268:223–284, 2006) and of Bleher and Liechty (Commun. Math. Phys., 286:777–801, 2009), in which the large n asymptotics is obtained for the partition function Zn of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.