A cold gas sample of metastable helium atoms ($2^3S_1$) is a possible candidate for observing Bose-Einstein condensation. Theoretical studies predict that Penning collisions, a major obstacle for achieving high density, are suppressed by several orders of magnitude due to spin polarization of the sample. This suppression should allow the transition to happen in the $\mu$K range. This polarization is obtained by magnetostatic trapping of the atoms. The high predicted value of the scattering length for helium in the $2^3S_1$ state may allow an efficient evaporative cooling in the magnetic trap. This thesis summarizes the theoretical predictions about elastic and inelastic collisions between metastable helium atoms and describes the apparatus leading to the loading of the magnetic trap. The technique used is based on laser cooling of an atomic beam assisted by Zeeman effect, followed by magneto-optical trapping and optical molasses. A first experimental limitation in the spatial density is encoutered during this stage. It is due to inelastic collisions enhanced by laser light. The pre-cooled sample is then transferred to the magnetic trap. Two experimental realizations of magnetostatic trapping are presented. The first one in a quadrupole is only a proof of principle. The second one is achieved in a Ioffe-Pritchard trap. We now have a magnetically trapped sample of $5\times 10^7$ atoms at a temperature of 200 $\mu$K, with a lifetime of 20 s.
