  We employ radio-frequency spectroscopy on weakly bound $^6$Li$_2$ molecules
to precisely determine the molecular binding energies and the energy splittings
between molecular states for different magnetic fields. These measurements
allow us to extract the interaction parameters of ultracold $^6$Li atoms based
on a multi-channel quantum scattering model. We determine the singlet and
triplet scattering lengths to be $a_s=45.167(8)a_0$ and $a_t=-2140(18)a_0$ (1
$a_0$ = 0.0529177 nm), and the positions of the broad Feshbach resonances in
the energetically lowest three $s-$wave scattering channels to be 83.41(15) mT,
69.04(5) mT, and 81.12(10) mT.
