In a mold for continuously casting a round billet with a curved type continuous casting apparatus, assuming that D0 (m) is an inner diameter at a lower mold edge and R0 (m) is a curvature radius of an outer curvature side at the lower mold edge, when a rate of change Tp (%/m) in mold inner diameter per unit length along a casting direction is expressed by Formula 1, and when a rate of change Rp (%/m) in curvature radius of an outer curvature side per unit length along the casting direction is expressed by Formula 2, the rate of change Tp in mold inner diameter and the rate of change Rp in curvature radius satisfy a relationship expressed by Formula 3; Tp=(1/D0)×(dD/dx)×100(%/m)  Formula 1 Rp=(1/R0)×(dR/dx)×100(%/m)  Formula 2 where D in Formula 1 is a mold inner diameter at a distance x away from an upper mold edge and R in Formula 2 is a curvature radius of the outer curvature side at the distance x, Rp=(Tp/2)×(D0/R0)  Formula 3 Uniform and good contact is obtained between the billet and a mold inner peripheral surface over a whole circumference, so that the casting-defect-free high-quality round billet can stably be produced.