We report element specific structural and magnetic investigations on Zn(1-x)Co(x)O epitaxial films using synchrotron radiation. Co dopants exclusively occupy Zn sites as revealed by x-ray linear dichroism having an unprecedented degree of structural perfection. Comparative magnetic field dependent measurements by x-ray magnetic circular dichroism and conventional magnetometry consistently show purely paramagnetic behavior for isolated Co dopant atoms with a magnetic moment of 4.8 (mu B). However, the total magnetization is reduced by approximately 30%, demonstrating that Co-O-Co pairs are antiferromagnetically coupled. We find no sign of intrinsic ferromagnetic interactions for isolated or paired Co dopant atoms in Co:ZnO films.