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The strong intervening absorption system at redshift 1.15 towards the verybright quasar HE 0515$-$4414 is the most studied absorber for measuringpossible cosmological variations in the fine-structure constant, $\alpha$. Weobserved HE 0515$-$4414 for 16.1$\,$h with the Very Large Telescope and presenthere the first constraint on relative variations in $\alpha$ withparts-per-million (ppm) precision from the new ESPRESSO spectrograph:$\Delta\alpha/\alpha = 1.3 \pm 1.3_{\rm stat} \pm 0.4_{\rm sys}\,{\rm ppm}$.The statistical uncertainty (1$\sigma$) is similar to the ensemble precision ofprevious large samples of absorbers, and derives from the high S/N achieved($\approx$105 per 0.4$\,$km$\,$s$^{-1}$ pixel). ESPRESSO's design, andcalibration of our observations with its laser frequency comb, effectivelyremoved wavelength calibration errors from our measurement. The high resolvingpower of our ESPRESSO spectrum ($R=145000$) enabled the identification of verynarrow components within the absorption profile, allowing a more robustanalysis of $\Delta\alpha/\alpha$. The evidence for the narrow components iscorroborated by their correspondence with previously detected molecularhydrogen and neutral carbon. The main remaining systematic errors arise fromambiguities in the absorption profile modelling, effects from redispersing theindividual quasar exposures, and convergence of the parameter estimationalgorithm. All analyses of the spectrum, including systematic error estimates,were initially blinded to avoid human biases. We make our reduced ESPRESSOspectrum of HE 0515$-$4414 publicly available for further analysis. Combiningour ESPRESSO result with 28 measurements, from other spectrographs, in whichwavelength calibration errors have been mitigated, yields a weighted mean$\Delta\alpha/\alpha = -0.5 \pm 0.5_{\rm stat} \pm 0.4_{\rm sys}\,$ppm atredshifts 0.6-2.4.

Fundamental physics with ESPRESSO: Precise limit on variations in the fine-structure constant towards the bright quasar HE 0515$-$4414