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This paper describes a design of a cryocooler-cooled 19 T superconducting magnet. The technical features of the magnet are a Bi2223 insert coil composed of 25 double pancake coils, Nb3Sn coils using a Nb3Sn wire reinforced with Nb-Ti-Cu compound, and a cooling structure using two types of cryocoolers. Coil protection from quenching was confirmed by numerical analysis. A preliminary experiment was carried out in order to investigate the influence of the bending strain upon a maximum permissible hoop stress of Ag-sheathed Bi2223 tape conductor.