We investigated the electronic and magnetic properties of thick films of molecular Cr8 and Cr7Ni antiferromagneticrings by means of x-ray absorption spectroscopy and x-ray magnetic circular dichroismXMCD. We determined the local symmetries, the electronic configuration, and the values of orbital and spinmoments at the Cr and Ni sites of the molecular rings. XMCD measurements show that the correlation betweenthe Cr and Ni spins in the Cr7Ni molecular ring switches from antiferromagnetic to ferromagnetic withincreasing temperature. Experimental data are interpreted using XMCD sum rules that allow the separateevaluation of the spin and the orbital contributions to the total magnetic moment of the ring as a function oftemperature and magnetic field. The magnetic behaviors experimentally observed are compared with the resultsof spin-Hamiltonian calculations, based on microscopic parameters derived by inelastic-neutron scattering andlow-temperature specific-heat measurements. The very good agreement between experimental data and calculationsis a clear indication of the integrity of molecules. The temperature dependence of the ion magneticmoments results from the interplay between Zeeman and isotropic-exchange contributions, and is well capturedby the theoretical model.