We have performed radio-frequency dissociation spectroscopy of weakly bound ^6Li_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 RF 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 ^6Li scattering potential using coupled-channel calculations. From this new potential, we determine the pole positions of the broad ^6Li Feshbach resonances with an accuracy better than 7 times 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. For example, 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).
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