Neutrinos in the cosmic ray flux with energies near 1 EeV and above are detectable with the Surface Detector array of the Pierre Auger Observatory. We report here on searches through Auger data from 1 January 2004 until 20 June 2013. No neutrino candidates were found, yielding a limit to the diffuse flux of ultra-high energy neutrinos that challenges the Waxman-Bahcall benchmark predictions. Neutrino identification is attempted using the broad time-structure of the signals expected in the SD stations, and is effciently done for neutrinos of all flavors interacting in the atmosphere at large zenith angles, as well as for "Earth-skimming" neutrino interactions in the case of tau neutrinos. In this paper the searches for downward-going neutrinos in the zenith angle bins 60°-75° and 75°-90° as well as for upward-going neutrinos, are combined to give a single limit. The 90% C.L. single-flavor limit to the diffuse flux of ultra-high energy neutrinos with an E^-2 spectrum in the energy range 1.0 x 10^17 eV - 2.5 x 10^19 eV is Enu^2 dNnu/dEnu < 6.4 x 10^-9 GeV cm^-2 s^-1 sr^-1.
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