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We propose an intrinsic nonlinear spin magnetoelectric effect in magnetic materials, offering the potential of all-electric control of spin degree of freedom in centrosymmetric magnets, which reside outside of the current paradigm based on linear spin response. We reveal the band geometric origin of this effect in the momentum and magnetization space Berry connection polarizabilities, and clarify its symmetry characters. As an intrinsic effect, it is determined solely by the material's band structure and represents a material characteristic. Combining our theory with first-principles calculations, we predict sizable nonlinear spin magnetoelectricity in single-layer MnBi$_{2}$Te$_{4}$, which can be detected in experiment. Our theory paves the way for exploring rich nonlinear spintronic effects and novel device concepts based on them.