We present measurements of a topological property, the Chern number (C_1), of a closed manifold in the space of two-level system Hamiltonians, where the two-level system is formed from a superconducting qubit. We manipulate the parameters of the Hamiltonian of the superconducting qubit along paths in the manifold and extract C_1 from the nonadiabitic response of the qubit. By adjusting the manifold such that a degeneracy in the Hamiltonian passes from inside to outside the manifold, we observe

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... topological transition C_1 = 1 → 0. Our measurement of C_1 is quantized to within 2 percent on either side of the transition.