  We simultaneously trap ultracold lithium and cesium atoms in an optical
dipole trap formed by the focus of a CO$_2$ laser and study the exchange of
thermal energy between the gases. The cesium gas, which is optically cooled to
$20 \mu$K, efficiently decreases the temperature of the lithium gas through
sympathetic cooling. The measured cross section for thermalizing
$^{133}$Cs-$^7$Li collisions is $8 \times 10^{-12}$ cm$^2$, for both species in
their lowest hyperfine ground state. Besides thermalization, we observe
evaporation of lithium purely through elastic cesium-lithium collisions
(sympathetic evaporation).
