Owing to the weak mutual interactions, spontaneous nuclear magnetic ordering in metallic copper and silver occurs at 60 nK and 560 pK, respectively. These extremely low spin temperatures can be reached by two-stage adiabatic nuclear demagnetization. Spin ordering has been investigated by employing magnetic susceptibility measurements on copper and silver and by using neutron diffraction techniques on copper. Three antiferromagnetic phases in the field-entropy plane have been discovered in copper, caused by competition between the dipolar and Ruderman-Kittel exchange interactions; only one ordered state has been found in silver. Negative spin temperatures have been produced in silver as well, and a clear ferromagnetic tendency was observed when T <0. The theoretically calculated spin-spin interactions, ordering temperatures, magnetic phase diagrams and ordered spin structures are in good overall agreement with experimental data for these two metals.