  The layered organic compounds, kappa-(BEDT-TTF)_2Cu[N(CN)_2X (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 (CESR) 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 hopping to the next molecular
layer. 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 inter-plane exchange field is comparable or less than the
typically 1 mT dipolar field and almost 10^6 times less than the intra-layer
exchange field.
