  We report on the formation of a stable quantum degenerate mixture of
fermionic $^6$Li and bosonic $^{133}$Cs in an optical trap by sympathetic
cooling near an interspecies Feshbach resonance. New regimes of the quantum
degenerate mixtures are identified. With moderate attractive interspecies
interactions, we show that a degenerate Fermi gas of Li can be fully confined
in the Cs condensate without external potentials. For stronger attraction where
mean-field collapse is expected, no such instability is observed. In this case,
we suggest the stability is a result of dynamic equilibrium, where the
interspecies three-body loss prevents the collapse. Our picture is supported by
a rate equation model, and the crossover between the thermalization rate and
the observed inelastic loss rate in the regime where the mean-field collapse is
expected to occur.
