About 99 per cent of solar energy is produced through sequences of nuclear reactions that convert hydrogen into helium, starting from the fusion of two protons (the pp chain). The neutrinos emitted by five of these reactions represent a unique probe of the Sun's internal working and, at the same time, offer an intense natural neutrino beam for fundamental physics. Here we report a complete study of the pp chain. We measure the neutrino-electron elastic-scattering rates for neutrinos produced by four reactions of the chain: the initial proton-proton fusion, the electron-capture decay of beryllium-7, the three-body proton-electron-proton (pep) fusion, here measured with the highest precision so far achieved, and the boron-8 beta decay, measured with the lowest energy threshold. We also set a limit on the neutrino flux produced by the He-3-proton fusion (hep). These measurements provide a direct determination of the relative intensity of the two primary terminations of the pp chain (pp-I and pp-II) and an indication that the temperature profile in the Sun is more compatible with solar models that assume high surface metallicity. We also determine the survival probability of solar electron neutrinos at different energies, thus probing simultaneously and with high precision the neutrino flavour-conversion paradigm, both in vacuum and in matter-dominated regimes.