  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 ($\Delta\alpha/\alpha$). 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 $\Delta\alpha/\alpha$ from a single
absorption system to date, $\Delta\alpha/\alpha=-1.42\pm0.55_{\rm
stat}\pm0.65_{\rm sys}$ 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 $\Delta\alpha/\alpha$ 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.
