The brightest southern quasar above redshift $z=1$, HE 0515$-$4414, with its strong intervening metal absorption-line system at $z_{abs}=1.1508$, provides a unique opportunity to precisely measure or limit relative variations in the fine-structure constant ($Deltaalpha/alpha$). A variation of just $sim$3 parts per million (ppm) would produce detectable velocity shifts between its many strong metal transitions. Using new and archival observations from the Ultraviolet and Visual Echelle Spectrograph (UVES) we obtain an extremely high signal-to-noise ratio spectrum (peaking at S/N $approx250$ pix$^{-1}$). This provides the most precise measurement of $Deltaalpha/alpha$ from a single absorption system to date, $Deltaalpha/alpha=-1.42pm0.55_{rm stat}pm0.65_{rm sys}$ ppm, comparable with the precision from previous, large samples of $sim$150 absorbers. The largest systematic error in all (but one) previous similar measurements, including the large samples, was long-range distortions in the wavelength calibration. These would add a $sim$2 ppm systematic error to our measurement and up to $sim$10 ppm to other measurements using Mg and Fe transitions. However, we corrected the UVES spectra using well-calibrated spectra of the same quasar from the High Accuracy Radial velocity Planet Searcher (HARPS), leaving a residual 0.59 ppm systematic uncertainty, the largest contribution to our total systematic error. A similar approach, using short observations on future, well-calibrated spectrographs to correct existing, high S/N spectra, would efficiently enable a large sample of reliable $Deltaalpha/alpha$ measurements. The high S/N UVES spectrum also provides insights into analysis difficulties, detector artifacts and systematic errors likely to arise from 25-40-m telescopes.
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