We describe a novel approach based on atomic force microscopy to determine the dihedral contact angle of a liquid polymer on top of a solid substrate. By cross-linking the polymer in an ion beam, the polymer surface becomes amenable to AFM imaging. This sample preparation enables us to take topography scans of the polymer in a solid state. The cross section of the contact area of a polymer drop and the substrate can then be utilized to determine the contact angle between polymer and substrate. Extremely low contact angles in the range 2-8 degrees are reproducibly measured. Control measurements carried out with optical phase modulated interference microscopy gave contact angles well comparable to the AFM results. We use our method in a study of polymer dewetting on top of a network of itself.