Understanding the quantum dynamics of strongly interacting fermions is a problem relevant to diverse forms of matter, including high-temperature superconductors, neutron stars, and quark-gluon plasma. An appealing benchmark is offered by cold atomic gases in the unitary limit of strong interactions. Here we study the dynamics of a transversely magnetized unitary Fermi gas in an inhomogeneous magnetic field. We observe the demagnetization of the gas, caused by diffusive spin transport. At low temperatures, the diffusion constant saturates to the conjectured quantum-mechanical lower bound \(\simeq \hbar/m\), where \(m\) is the particle mass. The development of pair correlations, indicating the transformation of the initially non-interacting gas towards a unitary spin mixture, is observed by measuring Tan's contact parameter.