We have studied the spontaneous antiferromagnetic (AF) order in the nuclear spin system of copper by use of neutron-diffraction experiments at nanokelvin temperatures. Copper is an ideal model system as a nearest-neighbor-dominated spin-3/2 fcc antiferromagnet. The phase diagram has been investigated by measuring the magnetic-field dependence of the (100) reflection, characteristic of a type-I AF structure, and of a Bragg peak at (0 2/3 2/3). The results suggest the presence of high-field (100) phases at 0.12 less-than-or-equal-to B less-than-or-equal-to B(c) almost-equal-to 0.26 mT, for B along either the  or  crystalline axes, intermediate-field (0 2/3 2/3) structures around B = 0.09 mT for all field directions, and a zero-field (100) phase. No reflection corresponding to a high-field phase for B parallel-to  has been found. The phase transition between the high-field phase and the intermediate-field structure is of first order. The change from (0 2/3 2/3) at intermediate fields to (100) at zero field is associated with a large region (0.02 less-than-or-equal-to B less-than-or-equal-to 0.06 mT) of coexisting-(100) and (0 2/3 2/3)-type Bragg peaks, and can be interpreted as either a two-phase region with a first-order transition at approximately 0.06 mT and huge hysteresis effects or as a single multiple-k phase which continuously transforms from being determined by (0 2/3 2/3) at approximately 0.1 mT and (100) at zero field. The neutron-diffraction data have been compared with results of earlier susceptibility measurements in order to identify the translational periods of the three previously found antiferromagnetic phases for B parallel-to . Recent theoretical work has yielded results in agreement with our experimental data.