  We present precision measurements with MHz uncertainty of the energy gap
between asymptotic and well bound levels in the electronic ground state X
$^1\Sigma_{\mathrm{g}}^+$ of the $^{39}$K$_2$ molecule. The molecules are
prepared in a highly collimated particle beam and are interrogated in a
$\Lambda$-type excitation scheme of optical transitions to long range levels
close to the asymptote of the ground state, using the electronically excited
state A $^1\Sigma^+_{\rm u}$ as intermediate one. The transition frequencies
are measured either by comparison with I$_2$ lines or by absolute measurements
using a fs-frequency comb. The determined level energies were used together
with Feshbach resonances from cold collisions of $^{39}$K and $^{40}$K reported
from other authors to fit new ground state potentials. Precise scattering
lengths are determined and tests of the validity of the Born-Oppenheimer
approximation for the description of cold collisions at this level of precision
are performed.
