Nuclear magnetism in metallic copper has been studied by demagnetizing highly polarized spins to low fields where spin-spin interactions dominate. In earlier experiments anomalous spin-lattice relaxation caused by impurities warmed up nuclear spins too fast; this adverse effect was overcome by selective oxidation of impurities. In zero field the critical temperature Tc of the antiferromagnetic transition is 58±10 nK, and during the first-order phase change the entropy increases from (0.48±0.03)ℛ ln 4 to (0.61±0.03)ℛ ln 4. The critical field Bc=0.27±0.01 mT. The entropy and the static susceptibility of the nuclear spins were measured as a function of temperature when B=0. These curves agree with theory in the paramagnetic state. In a polycrystalline sample two anomalies were observed at the lowest entropies in the NMR line shapes of the dynamic susceptibility and in the behavior of the static susceptibility. However, when measuring the static susceptibility of a single-crystal specimen in the three Cartesian directions, three different ordered phases were found. These antiferromagnetic states are described and the B-S phase diagram is presented. Metastability and nonadiabaticity are discussed. The observed large reduction of Tc from the mean field calculation TMF=230 nK is caused by fluctuations. The free electron model of the Ruderman-Kittel (RK) interaction seems to be able to explain only one ordered phase. However, relatively small changes to the RK range function or inclusion of non-s-electron-mediated interactions to the Hamiltonian may increase the number of ordered phases to three. Long-living metastable states are another possible explanation for the observations.