We have produced a very cold sample of spin-polarized trapped atoms. The technique used dramatically simplifies the production of laser-cooled atoms. In this experiment, 1.8\ifmmode\times\else\texttimes\fi{}${10}^{7}$ neutral cesium atoms were optically captured directly from a low-pressure vapor in a small glass cell. We then cooled the 1-${\mathrm{mm}}^{3}$ cloud of trapped atoms and loaded it into a low-field magnetic trap in the same cell. The magnetically trapped atoms had an effective temperature as low as 1.1\ifmmode\pm\else\textpm\fi{}0.2 \ensuremath{\mu}K, which is the lowest kinetic temperature ever observed and far colder than any previous sample of trapped atoms.