The change of the interface kinetic coefficient with the interface temperature under the non-equilibrium solidification condition is considered in the growth model of a spherical crystal. The first-order approximation solutions of temperature and interface for the spherical crystal growth in the undercooled melt are obtained by the asymptotic analysis method. The effects of the nonlinear interface kinetic undercooling on the interface morphology and the growth velocity of the spherical crystal in the undercooled melt are studied. The results show that as the interface kinetic coefficient increases, the growth velocity of the spherical crystal increases; as the interface kinetic coefficient decreases, the growth velocity of the spherical crystal decreases. Compared with the situation of neglecting interface kinetics, the nonlinear interface kinetics during crystal growth significantly decreases the growth velocity of the spherical crystal.