The adsorption of a long weakly charged flexible polyelectrolyte in a salt solution onto an oppositely charged spherical surface is investigated. An analytical solution for Green's function is derived, which is valid for any sphere radius and consistently recovers the result of a planar surface in the limit of large sphere radii, by substituting the Debye-Huckel potential via the Hulthen potential. Expressions for critical quantities like the critical radius and the critical surface charge density are provided. In particular, we find a universal critical line for the sphere radius as a function of the screening length separating adsorbed from desorbed states. Moreover, results for the monomer density distribution, adsorbed layer thickness, and the radius of gyration are presented. A comparison of our theoretical results with experiments and computer simulations yields remarkably good agreement.