The lattice-gas approach is generalized to incorporate features of the configurational problem posed by the randomly hydrogen-bonded gel model for liquid water. Because it possesses sublattices characterized by tetrahedral angles associated with triads of sites, a body-centered cubic (bcc) lattice is used. Each water molecule is allowed 12 orientations with respect to the bcc lattice. When two nearest neighbors have relative orientations which permit hydrogen bonding, they are assigned a hydrogen bond energy. When hydrogen bonding is not permitted the pair is assigned one of two weaker interaction energies. Like the simple lattice gas, this model displays a vapor-liquid phase transition. The critical site density proves to be less than 1/2. The model should also exhibit a transition to a solid phase as a result of the possibility of complete hydrogen bonding associated with exclusive occupation of one sublattice. Excellent agreement is obtained with the observed temperature dependence of the second virial coefficient. The agreement in the case of the third virial coefficient is poor, however. The mean field approximation is shown to be inadequate for quantitative description of the vapor-liquid transition and the properties of the liquid phase.