  For calculating low-energy properties of a dilute gas of atoms interacting
via a Feshbach resonance, we develop an effective theory in which the
parameters that enter are an atom-molecule coupling strength and the magnetic
moment of the molecular resonance. We demonstrate that for resonances in the
fermionic systems $^{6}$Li and $^{40}$K that are under experimental
investigation, the coupling is so strong that many-body effects are appreciable
even when the resonance lies at an energy large compared with the Fermi energy.
We calculate a number of many-body effects, including the effective mass and
the lifetime of atomic quasiparticles in the gas.
