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We measure the spin splitting in a magnetic field $B$ of localized states insingle-electron transistors using a new method, inelastic spin-flipcotunneling. Because it involves only internal excitations, this techniquegives the most precise value of the Zeeman energy $\Delta = \ZeemanE$. In thesame devices we also measure the splitting with $B$ of the Kondo peak indifferential conductance. The Kondo splitting appears only above a thresholdfield as predicted by theory. However, the magnitude of the Kondo splitting athigh fields exceeds $2 \ZeemanE$ in disagreement with theory.Comment: Replaced with version published in Phys. Rev. Let

Measurements of Kondo and Spin Splitting in Single-Electron Transistors