We use an all-optical trap to confine a strongly attractive two-state mixture of lithium fermions. By measuring the rate of evaporation from the trap, we determine the effective elastic scattering cross section 4pia(2) to show that the magnitude of the scattering length |a| is very large, in agreement with predictions. We show that the mixture is stable against inelastic decay provided that a small bias magnetic field is applied. For this system, the s-wave interaction is widely tunable at low magnetic field, and can be turned on and off rapidly via a Raman pi pulse. Hence, this mixture is well suited for fundamental studies of an interacting Fermi gas.