  We compute the genus $0$ free energy for the $2$-matrix model with quartic
interactions, which acts as a generating function for the Ising model's
partition function on a random, $4$-regular, planar graph. This rigorously
confirms the predictions of V.A. Kazakov and D.V. Boulatov on this model, and
provides a new parametric formula for the free energy. We also give a
characterization of the phase space of the model. Our analysis is based on a
steepest descent Riemann-Hilbert analysis of the associated biorthogonal
polynomials and the corresponding isomonodromic $\tau$-function. A key
ingredient in the analysis is a parametrization of the spectral curve.
