For the transmission of electrical current, determined superconducting current carriers are accommodated in a cryostat, in which they are cooled with an undercooled cryogenic cooling medium, e.g. liquid nitrogen. The current carrier is electrically connected at the ends thereof to two normally conducting current supply means. The cooling medium is undercooled from a storage container to a temperature below its boiling temperature and supplied to the cryostat via a cooling medium inlet, brought into thermal contact with the superconducting current carrier, and subsequently discharged via a cooling medium outlet of the cryostat. According to the invention, the cooling medium from the cryostat is to be used for cooling at least one of the normally conducting current supply means.