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).
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