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We present precise, subwavelength optical intensity measurements using a single trapped ^{87}Rb atom as a sensor. The intensity is measured by the scalar ac Stark shift it produces on the F=1→F^{′}=2 hyperfine transition of the D_{2} line, chosen for its F^{′}=F+1 structure and very small tensor polarizability. To boost signal and reduce measurement-induced perturbations, we use a quantum jump spectroscopy technique in which a single absorbed photon on a transition of interest induces the scattering of hundreds of photons on a bright closed transition. The method greatly reduces systematic effects associated with the atomic state, optical polarization, probe power, and atom heating, and gives the atomic temperature as a second spectroscopic observable. We demonstrate the method by measuring the intensity at the focus of an optical tweezer.