A new model is proposed to account for the long-range hydrophobic attraction repeatedly observed for thin water films between two stable (solid) hydrophobic surfaces. The model is based on the notion of structurally organized, elongated water clusters that span the gap between the hydrophobic surfaces. Two features are noted: (i) Mixing entropy due to the mixing of the clusters and the remainder of the water in the thin film is explicitly taken into account. (ii) A term is invoked that depends inversely on the film thickness, which accounts for the free-energy change associated with reorganizing the film as the film thickness varies. Fitting to experimental surface force data resulted in parameter values of reasonable magnitudes. The model developed covers film thicknesses from about 2 nm and above. On this basis, the amazingly long range of the hydrophobic attraction can be attributed to the formation of bridging, quasi-cylindrical clusters having a radius on the order of 1 to 2 nm.