The invention relates to a method for the continuous production of unsaturated carboxylic acid anhydrides of general formula (I): R—C(O)—O—C(O)—R (I), wherein R is an unsaturated organic group with 2 to 12 C atoms, by reaction of a ketene with an unsaturated carboxylic acid of general formula (II): R—COOH (II), wherein R is defined as above, in an apparatus which has a reaction zone (1) for the reaction of a ketene with an unsaturated carboxylic acid of general formula (II), and a reaction zone (2) for the subsequent reaction of the crude anhydride mixture and a rectification column with an upper, a middle and a lower zone, in the sump of which an inert boiling oil is present.