We have measured the elastic collision cross section for spin polarized atomic cesium. Neutral cesium atoms are optically cooled, then loaded into a dc magnetic trap. We infer the scattering rate from the rate at which anisotropics in the initial energy distribution are observed to relax. The cross section for F = 3, mf = -3 on ^^3, w/r = -3 is 1.5(4) x 10~1 2 cm 2 , and is independent of temperature from 30 to 250 ^K. This determination clarifies the technical requirements for attaining Bose-Einstein condensation in a magnetically trapped Cs vapor. We also study heating due to glancing collisions with 300 K background Cs atoms.PACS numbers: 34.40.+ n, 05.30Jp, 32.80.Pj, 34.50.-s Recent advances in optical trapping and cooling techniques have allowed collision studies in an entirely new regime of temperatures. In this regime of mK or even colder temperatures, collisions take on unusual characteristics, and often have cross sections many orders of magnitude different from that observed at previously attainable temperatures. Studies of ultralow-temperature collisions have investigated several inelastic processes which cause loss from optical traps: two-photon quasimolecular photoionization [1], photon-assisted collisions [2,3], and hyperfine-state changing collisions [2,4]. All of these processes are inelastic, and while they involve low initial energies for the atoms involved, the final states have kinetic energies of 1 K or greater. It appears that this allows these processes to be treated semiclassically, although the exact accuracy of such treatments is currently a topic of debate. In the work presented here, we examine elastic collisions in which the initial and final energies are both much less than 1 mK. In order to study such low-energy collisions we have developed techniques for observing collisions in very cold samples of magnetically trapped atoms. From these measurements, we deduce the s-wave collision cross section.The elastic scattering cross section is of particular interest because of its importance in determining the experimental feasibility of achieving Bose-Einstein condensation (BEC) in a cesium vapor. As demonstrated in experiments with spin polarized hydrogen [5], evaporative cooling of magnetically trapped atoms is a promising strategy for achieving the temperature and density necessary for the BEC phase transition, because evaporative cooling does not have the density and temperature limitations encountered in optical cooling and trapping [2,6].To study the very low energy elastic collisions, we first accumulate Cs atoms in a Zeeman-shift optical trap (ZOT) in a low pressure cell [7]. The atoms are then optically cooled and spin polarized in the F = 3, m/r = -3 state in preparation for magnetic trapping. After all laser light is shut off, magnetic fields are turned on around the atoms in situ [7], thereby trapping them.In this work we use a type of dc magnetic trap which has been previously proposed but, to our knowledge, never used to trap neutral atoms. The field coils consist of wire woun...