The Surface Detector of the Pierre Auger Observatory is sensitive to neutrinos of all flavours above 0.1 EeV. These interact through charged and neutral currents in the atmosphere giving rise to extensive air showers. When interacting deeply in the atmosphere at nearly horizontal incidence, neutrinos can be distinguished from regular hadronic cosmic rays by the broad time structure of their shower signals in the water-Cherenkov detectors. In this paper we present for the first time an analysis based on down-going neutrinos. We describe the search procedure, the possible sources of background, the method to compute the exposure and the associated systematic uncertainties. No candidate neutrinos have been found in data collected from 1 January 2004 to 31 May 2010. Assuming an E^-2 differential energy spectrum the limit on the single flavour neutrino is (E^2 * dN/dE) < 1.74x10^-7 GeV cm^-2 s^-1 sr^-1 at 90% C.L. in the energy range 1x10^17 eV < E < 1x10^20 eV.