  In this paper, properties of a homogeneous Bose gas with a Feshbach resonance
are studied in the dilute region at zero temperature. The stationary state
contains condensations of atoms and molecules. The ratio of the molecule
density to the atom density is $\pi na^3$. There are two types of excitations,
molecular excitations and atomic excitations. Atomic excitations are gapless,
consistent with the traditional theory of a dilute Bose gas. The molecular
excitation energy is finite in the long wavelength limit as observed in recent
experiments on $^{85}$Rb. In addition, the decay process of the condensate is
studied. The coefficient of the three-body recombination rate is about 140
times larger than that of a Bose gas without a Feshbach resonance, in
reasonably good agreement with the experiment on $^{23}$Na.
