We have constructed a two-stage nuclear demagnetization cryostat for neutron diffraction studies of nuclear magnetism in copper. The cryostat is combined with a two-axis neutron spectrometer which can use both polarized and unpolarized neutrons. By demagnetizing highly polarized copper nuclear spins, the nuclei could be cooled below the ordering temperatureT N≃60 nK, while keeping the lattice at a considerably higher temperature between 50 and 100 µK. The neutron beam increases the lattice temperature in the sample by a factor of two or more, thereby considerably shortening the time for measurements in the ordered state; both our calculations and the experiments yield 1 nW beam heating. Polarized neutron experiments show that the scattered intensities from the strong fcc reflections are severely reduced by extinction. This makes the sample not very suitable for further studies with polarized neutrons. By observing the (100) Bragg reflection, we have unambiguously proven antiferromagnetic ordering of the copper nuclear spins. Using a linear, position-sensitive detector, the time evolution of this peak was followed during the warm-up of the nuclear spin system. The peak intensity was found to depend strongly on the external magnetic field between zero and the critical fieldB c=0.25 mT, indicating the existence of at least two antiferromagnetic phases. The results are compared to previous measurements of the magnetic susceptibility. Theoretical calculations do not provide a full explanation for our experimental data.