  We report on the creation of a degenerate Fermi gas consisting of a balanced
mixture of atoms in three different hyperfine states of $^6$Li. This new system
consists of three distinguishable Fermions with different and tunable
interparticle scattering lengths $a_{12}$, $a_{13}$ and $a_{23}$. We are able
to prepare samples containing $5 \cdot 10^4$ atoms in each state at a
temperature of about $215 $nK, which corresponds to $T/T_F \approx 0.37$. We
investigated the collisional stability of the gas for magnetic fields between 0
and 600 G and found a prominent loss feature at 130 G. From lifetime
measurements we determined three-body loss coefficients, which vary over nearly
three orders of magnitude.
