In the continuous casting of round or polygonal billet and bloom formats, use is made of moulds the mould cavity of which comprises a copper tube (3) which is intensively cooled by means of water-circulation cooling. In order to increase the cooling capacity on the one hand and the dimensional stability of the mould cavity (4) on the other hand, and also extend the total service life of the copper tube (3), it is proposed to provide the copper tube (3) with a supporting shell (12) or supporting plates over the entire circumference at the tube outer lateral surface (5). For the cooling of the copper tube (3), cooling ducts (6) for guiding the cooling water are arranged on the copper tube (3) or on the supporting shell (12). The cooling ducts (6) are distributed over the entire circumference at the tube outer lateral surface (5) and extend substantially over the entire mould length.