Abstract : This thesis presents the construction of an experiment towards Fermi degeneracy with $^6$Li evaporatively cooled in a magnetic trap. It also describes theoretically some properties of a Fermi gas in the degenerate quantum regime. Due to the Pauli exclusion principle, the standard evaporative cooling of a Fermi gas is not effective at low temperatures. The method proposed here avoids this limitation by using a mixture of fermions ($^6$Li) and bosons ($^7$Li). The setup consists of a laser-slowed atomic beam of lithium, a magneto-optical trap with large diameter laser beams, and a magnetic trap designed to give high magnetic confinement. A new semiconductor laser source delivering many frequency components at 670 nm has been developed. With this source we trapped 4 10$^(8)$ atoms of $^6$Li and 10$^(10)$ atoms of $^7$Li simultaneously in a magneto-optical trap at a temperature of 1 mK. The atoms are then transferred a distance of 5çm into a high confining Ioffe-Pritchard trap. This trap contains $4\,10^8$ atoms of $^7$Li at a temperature of $\simeq 2$ mK with a lifetime of 75 s. In addition , this thesis proposes a new method to measure the Fermi degeneracy of an ideal gas. This method, which is very sensitive, is based on the inhibition of collisional relaxation of a gas of test particles with a Fermi gas cooled below the Fermi temperature. Finally we propose a new method for laser cooling which combines inelastic collisions and optical pumping.