A theory of bipolaron states in a spherical parabolic potential well is developed applying the Feynman variational principle. The basic parameters of the bipolaron ground state (the binding energy, the number of phonons in the bipolaron cloud, and the bipolaron radius) are studied as functions of the radius R of the potential well. Analytical expressions for bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at R>>1 (where R is expressed in units of the polaron radius), the influence of confinement on the bipolaron binding energy W(R) is described by the function ~1/R², while at small sizes this influence is more complicated: W(R) passes through a maximum in the region R<1.