The chemical evolution of neutron capture elements in the Milky Way is still a matter of debate. Although more and more studies investigate their chemical behaviour, there is still a lack of a significant large sample of abundances of a key heavy element: lead. Lead is the final product of the s-process nucleosynthesis channel and is one of the most stable heavy elements. The goal of this article is to present the largest catalogue of homogeneous Pb abundances, in particular for metallicities higher than -1.0dex, and then to study the lead content of the Milky Way. We analysed high-resolution spectra from the ESO UVES and FEROS archives. Atmospheric parameters were taken from the AMBRE parametrisation. We used the automated abundance method GAUGUIN to derive LTE and NLTE lead abundances in 653 slow-rotating FGK-type stars from the 368.34nm PbI line, with metallicities ranging from -2.9 to 0.6 ex Within this sample, no lead-enhanced Asymptotic Giant Branch (AGB) stars were found, but nine lead-enhanced metal-poor stars ([Pb/Fe]>1.5) were detected. Most of them were already identified as carbon-enhanced metal-poor stars with enrichments in other s-process species. The lead abundance of 13 Gaia Benchmark Stars are also provided. We then investigated the Pb content of the Milky Way disc by computing vertical and radial gradients and found a slightly decreasing [Pb/Fe] radial trend with metallicity. This trend together with other related ratios ([Pb/Eu], [Pb/Ba], and [Pb/alpha]) are interpreted thanks to chemical evolution models. The two-infall model closely reproduces the observed trends with respect to the metallicity. It is also found that the AGB contribution to the Pb Galactic enrichment has to be strongly reduced. Moreover, the contribution of massive stars with rather high rotational velocities should be favoured in the low-metallicity regime.