The method and the apparatus are used for the low-temperature fractionation of a fluid mixture. The fluid mixture is introduced into a separation column. At least a portion of the sump liquid from the separation column is introduced into a sump evaporator and at least partly evaporated there. At least a portion of the vapor produced in the sump evaporator is led back into a lower section of the separation column. A top product is removed at the top of the separation column, and a sump product is removed from the sump of the separation column or from the sump evaporator. The sump evaporator is operated by means of inductive heating.