We perform radio-frequency dissociation spectroscopy of weakly bound Li-6(2) Feshbach molecules using low-density samples of about 30 molecules in an optical dipole trap. Combined with a high magnetic field stability, this allows us to resolve the discrete trap levels in the radio-frequency dissociation spectra. This novel technique allows the binding energy of Feshbach molecules to be determined with unprecedented precision. We use these measurements as an input for a fit to the Li-6 scattering potential using coupled-channel calculations. From this new potential, we determine the pole positions of the broad Li-6 Feshbach resonances with an accuracy better than 7 x 10(-4) of the resonance widths. This eliminates the dominant uncertainty for current precision measurements of the equation of state of strongly interacting Fermi gases. As an important consequence, our results imply a corrected value for the Bertsch parameter xi measured by Ku et al. [Science 335, 563 (2012)], which is xi = 0.370(5)(8). DOI: 10.1103/PhysRevLett.110.135301