The layered organic compounds, kappa-(BEDT-TTF)2Cu[N(CN)2]X X=Cl, Br) are metals at ambient temperatures. At low temperatures, the Cl compound is a weakly ferromagnetic Mott insulator while the isostructural Br compound is a superconductor. We find by conduction electron spin resonance and antiferromagnetic resonance (AFMR) an extreme anisotropy of spin transport and magnetic interactions in these materials. In the metallic state spin diffusion is confined to single molecular layers within the spin lifetime of 10(-9) s. Electrons diffuse several hundreds of nm without interlayer hopping. In the magnetically ordered insulating phase of the Cl compound we observe and calculate the four AFMR modes of the weakly coupled single molecular layers. The interplane exchange field is comparable or less than the typically 1 mT dipolar field and almost 10(6) times less than the intralayer exchange field.