A fiber reinforced composite member molding apparatus includes a pair of molds for clamping prepreg formed of long carbon fibers impregnated with resin, induction heating coils for heating thermoplastic resin contained in the prepreg via the molds, and cooling passages for cooling the resin via the molds after the resin is melted, wherein the molds each have a design surface brought into contact with the layered prereg, the design surface being divided into a plurality of regions, and a plurality of cells provided along the design surface to be open at the back of the design surface and individually correspond to the regions of the design surface, the induction heating coils are arranged in the cells, and the cooling passages are formed in each of the molds to run along the design surface.