The exact series solution for the reaction rate of a spherical sink confined within a sphere is presented from reaction- to diffusion-limited condition based on the bispherical coordinate method. The reaction rate varies with the particle location and the size ratio of particle to enveloping sphere. The maximum and minimum rates take place when the particle and the confining sphere are at contact and concentric, respectively. A thin-gap analysis is employed to derive the rate expression analytically for a near contact case. While the reaction rate at near contact diverges logarithmically for a purely diffusion-limited condition, it remains finite for a fast surface reaction with finite rate constant. The average reaction rate is then calculated based on a prescribed particle distribution function. It is found that for a uniform distribution, the mean rate is at most 50% higher than that for the concentric case. Other than the hard-body interaction, additional attractive and repulsive interactions will enhance and reduce the mean rate, respectively.