  Correlation functions of the six and nineteen vertex models on an N \times N
lattice with domain wall boundary conditions are studied. The general
expression of the boundary correlation functions is obtained for the six vertex
model by use of the quantum inverse scattering method. The correlation
functions which are not "boundary" can be expressed as a linear sum of the
boundary correlation functions. For the nineteen vertex model, the boundary
correlation functions are shown to be expressed in terms of those for the six
vertex model.
