The subject of the present study is the dynamics of a single cavitation bubble in a spherical liquid cell surrounded by an infinite elastic solid. It is shown that volume confinement strongly affects the manifestation of the classical cavitation Blake threshold. In particular, at liquid cell sizes smaller than some critical size, the cavitation is completely suppressed by volumetric confinement. The system of equations for the dynamics of a confined bubble, accounting for the mass of gas in the bubble, surface tension, liquid compressibility, solid elasticity, and damping due to viscosity in the liquid cell, is derived. The pressure in the solid far away from the bubble is used as an external driving force. Linear analysis of the bubble dynamics, including consideration of the natural frequency and amplitude-phase frequency response of the bubble-in-cell system, is conducted. In the nonlinear case, bifurcation diagrams are considered to determine the dynamic response of small and large bubbles in the states below and above the cavitation threshold, respectively.