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We present a theory of a single-electron transistor exchange-coupled to a localized spin. We show how to gain detailed quantitative knowledge about the attached spin such as spin size, exchange coupling strength, Landé g-factor, and spin decay time $T_1$ by exploring the nonlinear DC current and noise in a constant magnetic field. Our studies are of particular relevance to spin-resolved scanning single-electron transistor microscopy, electronic transport through nanomagnets, and the effect of hyperfine interaction on transport electrons by surrounding nuclear spins.