The classical theories of shells and curved beams are used to develop the equations of motion of an elastically isotropic spherical shell attached to an elastic equatorial beam of rectangular cross section. The mass densities and elasticities of the shell and beam are, in general, different. Remarkably, for the natural frequencies, the final set of eight linear homogeneous algebraic equations uncouples into two sets of four. The only approximations made are of the same order of magnitude as those inherent in classical shell and beam theory. Although solutions of the shell equations involve Legendre functions (and not polynomials), the final set of algebraic equations involve only trigonometric and gamma functions. Several special exact solutions are given. In Part II, perturbation techniques are used to find the natural frequencies for beam-shell configurations ranging from nearly pure beams to nearly pure shells.