Context.The discovery of Proxima b marked one of the most important milestones in exoplanetary science in recent years. Yet the limited precision of the available radial velocity data and the difficulty in modelling the stellar activity calls for a confirmation of the Earth-mass planet.Aims.We aim to confirm the presence of Proxima b using independent measurements obtained with the new ESPRESSO spectrograph, and refine the planetary parameters taking advantage of its improved precision.Methods.We analysed 63 spectroscopic ESPRESSO observations of Proxima (Gl 551) taken during 2019. We obtained radial velocity measurements with a typical radial velocity photon noise of 26 cm s−1. We combined these data with archival spectroscopic observations and newly obtained photometric measurements to model the stellar activity signals and disentangle them from planetary signals in the radial velocity (RV) data. We ran a joint Markov chain Monte Carlo analysis on the time series of the RV and full width half maximum of the cross-correlation function to model the planetary and stellar signals present in the data, applying Gaussian process regression to deal with the stellar activity signals.Results.We confirm the presence of Proxima b independently in the ESPRESSO data and in the combined ESPRESSO+ HARPS+UVES dataset. The ESPRESSO data on its own shows Proxima b at a period of 11.218 ± 0.029 days, with a minimum mass of 1.29 ± 0.13M⊕. In the combined dataset we measure a period of 11.18427 ± 0.00070 days with a minimum mass of 1.173 ± 0.086M⊕. We get a clear measurement of the stellar rotation period (87 ± 12 d) and its induced RV signal, but no evidence of stellar activity as a potential cause for the 11.2 days signal. We find some evidence for the presence of a second short-period signal, at 5.15 days with a semi-amplitude of only 40 cm s−1. If caused by a planetary companion, it would correspond to a minimum mass of 0.29 ± 0.08M⊕. We find that forthe case of Proxima, the full width half maximum of the cross-correlation function can be used as a proxy for the brightness changes and that its gradient with time can be used to successfully detrend the RV data from part of the influence of stellar activity. The activity-induced RV signal in the ESPRESSO data shows a trend in amplitude towards redder wavelengths. Velocities measured using the red end of the spectrograph are less affected by activity, suggesting that the stellar activity is spot dominated. This could be used to create differential RVs that are activity dominated and can be used to disentangle activity-induced and planetary-induced signals. The data collected excludes the presence of extra companions with masses above 0.6M⊕at periods shorter than 50 days.