The first simulation study of the percolation of hydrogen bonded water clusters in the vicinity of the region of immiscibility of an aqueous solution (of tetrahydrofuran) is reported. Percolation of water is found in a wide concentration range on both sides of the liquid-liquid coexistence curve. An infinite cluster appears with a probability of 50% at a water fraction significantly lower than the one corresponding to the organic-rich branch of the coexistence curve. The fractal dimension d(f) of the infinite clusters at this threshold is found close to the two-dimensional (2D) value, d(f)(2D)congruent to1.9. Three-dimensional (3D) percolation clusters form at the organic-rich branch of the coexistence curve. At this water concentration the fractal dimension of an infinite cluster reaches the 3D value d(f)(3D)congruent to2.5 and the cluster size distribution follows a power law with an exponent taucongruent to2.2. The analysis of the clustering of the organic (tetrahydrofuran) molecules indicates that the immiscibility gap of an aqueous solution corresponds to the concentration interval where both components are above their respective percolation threshold.