An exact three-dimensional elastodynamic analysis for describing the acoustic resonance frequencies of an acoustic eccentric hollow sphere is derived. The Neumann boundary conditions for inner and outer sphere are considered. The translational addition theorem for spherical vector wave functions is employed to enforcing Neumann boundary conditions. The frequency equations in form of exact determinantal equations involving spherical Bessel functions and Wigner 3j symbols are obtained. Due to geometric symmetry for spherical cavity with inner concentric sphere, multiple degenerate acoustic resonance frequencies are occurred. According to the geometry parameters and frequency number, introduction of eccentricity has a different effect on the acoustic resonance frequency shift. Extensive numerical results have been carried out for acoustic resonance frequency of selected inner-outer radii ratios in a wide range of cavity eccentricities. The numerical results describe the imperative influence of cavity eccentricity and radii ratio on the resonance frequency of the acoustic hollow sphere. Some phenomena such as diminishing degenerate resonance frequency, increasing in the number of resonant frequencies through the splitting of degenerate modes and exchanging the mode of resonance frequencies are demonstrated and discussed.