  Since the problem of the residual entropy of square ice was exactly solved,
exact solutions for two-dimensional realistic ice models have been of interest.
In this paper, we study the exact residual entropy of ice hexagonal monolayer
in two cases. In the case that the external electric field along the z-axis
exists, we map the hydrogen configurations into the spin configurations of the
Ising model on the Kagom\'e lattice. By taking the low temperature limit of the
Ising model, we derive the exact residual entropy, which agrees with the result
determined previously from the dimer model on the honeycomb lattice. In another
case that the ice hexagonal monolayer is under the periodic boundary conditions
in the cubic ice lattice, we employ the six-vertex model on the square lattice
to represent the hydrogen configurations obeying the ice rules. The exact
residual entropy in this case is obtained from the solution of the equivalent
six-vertex model. Our work provides more examples of the exactly soluble
two-dimensional models.
