  We report the experimental evidence of the sympathetic cooling of $^{39}$K
with $^{87}$Rb down to 1 $\mu$K, obtained in a novel tight confining magnetic
trap. This allowed us to perform the first direct measurement of the elastic
cross section of $^{39}$K below 50 $\mu$K. The result obtained for the triplet
scattering length, $a_T = -51(7)$ Bohr radii, agrees with previous results
derived from photoassociation spectra and from Feshbach spectroscopy of
$^{40}$K.
