  The strong intervening absorption system at redshift 1.15 towards the very
bright quasar HE 0515$-$4414 is the most studied absorber for measuring
possible cosmological variations in the fine-structure constant, $\alpha$. We
observed HE 0515$-$4414 for 16.1$\,$h with the Very Large Telescope and present
here the first constraint on relative variations in $\alpha$ with
parts-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 of
previous large samples of absorbers, and derives from the high S/N achieved
($\approx$105 per 0.4$\,$km$\,$s$^{-1}$ pixel). ESPRESSO's design, and
calibration of our observations with its laser frequency comb, effectively
removed wavelength calibration errors from our measurement. The high resolving
power of our ESPRESSO spectrum ($R=145000$) enabled the identification of very
narrow components within the absorption profile, allowing a more robust
analysis of $\Delta\alpha/\alpha$. The evidence for the narrow components is
corroborated by their correspondence with previously detected molecular
hydrogen and neutral carbon. The main remaining systematic errors arise from
ambiguities in the absorption profile modelling, effects from redispersing the
individual quasar exposures, and convergence of the parameter estimation
algorithm. All analyses of the spectrum, including systematic error estimates,
were initially blinded to avoid human biases. We make our reduced ESPRESSO
spectrum of HE 0515$-$4414 publicly available for further analysis. Combining
our ESPRESSO result with 28 measurements, from other spectrographs, in which
wavelength 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 at
redshifts 0.6-2.4.
