Cold inelastic collisions between confined cesium (Cs) atoms and
Cs2 molecules are investigated inside a
CO2 laser dipole trap. Inelastic atom-molecule collisions can be observed and measured with a rate coefficient of $\ensuremath{\sim}2.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\text{ }\text{ }{\mathrm{cm}}^{3}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$, mainly independent of the molecular rovibrational state populated. Lifetimes of purely atomic and molecular samples are essentially limited by rest gas collisions. The pure molecular trap lifetime ranges 0.3--1 s, 4 times smaller than the atomic one, as is also observed in a pure magnetic trap. We give an estimation of the inelastic molecule-molecule collision rate to be $\ensuremath{\sim}{10}^{\ensuremath{-}11}\text{ }\text{ }{\mathrm{cm}}^{3}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$.