The electronic structure of the paramagnetic Ag4+ and Ag43+ clusters is investigated by the Dirac scattered-wave method. The results show that relativistic effects are not very large, but they cannot be neglected. Due to relativity, nonnegligible 4d-5s hybridization arises in the bonding molecular orbitals. Spin-orbit interaction splits the occupied valence d band by 0.9 eV. Molecular hyperfine interactions (hfi) are calculated through a first-order perturbation to the Dirac Hamiltonian. The predicted hfi and excitation energies have enabled us to distinguish clearly between Ag4+ and Ag43+. The hyperfine tensors for Ag4+ are predicted to be anisotropic, whereas those for Ag43+ are mostly isotropic. The theoretical spectral assignments for Ag43+ are in good agreement with recent optical and ESR studies.