The metal-weak tail of the metallicity distribution function (MDF) of the Galactic Halo stars contains crucial information on the formation mode of the first generation of stars. To determine this observationally, it is necessary to observe large numbers of extremely metal-poor stars. We present here the Turn-Off Primordial Stars survey (TOPoS) that is conducted as an ESO Large Programme at the VLT. This project has {four} main goals: (i) to understand the formation of low-mass stars in a low-metallicity gas: determine the metal-weak tail of the halo MDF below [M/H]=-3.5. In particular, we aim at determining the critical metallicity, that is the lowest metallicity sufficient for the formation of low-mass stars; (ii) to determine the relative abundance of the elements in extremely metal-poor stars, which are the signature of the massive first stars; (iii) to determine the trend of the lithium abundance at the time when the Galaxy formed; and (iv) to derive the fraction of C-enhanced extremely metal-poor stars with respect to normal extremely metal-poor stars. The large number of stars observed in the SDSS provides a good sample of candidates of stars at extremely low metallicity. Candidates with turn-off colours down to magnitude g=20 were selected from the low-resolution spectra of SDSS by means of an automated procedure. X-Shooter has the potential of performing the necessary follow-up spectroscopy, providing accurate metallicities and abundance ratios for several key elements for these stars. We here present the stellar parameters of the first set of stars. The nineteen stars range in iron abundance between -4.1 and -2.9 dex relative to the Sun. Two stars have a high radial velocity and, according to our estimate of their kinematics, appear to be marginally bound to the Galaxy and are possibly accreted from another galaxy.