Data collected at the Pierre Auger Observatory are used to establish an upper limit on the diffuse flux of tau neutrinos in the cosmic radiation. Earth-skimming tau neutrinos may interact in the Earth's crust and produce a tau lepton by means of charged-current interactions. The tau lepton may emerge from the Earth and decay in the atmosphere to produce a nearly horizontal shower with a typical signature, a persistent electromagnetic component even at very large atmospheric depths. The search procedure to select events induced by tau decays against the background of normal showers induced by cosmic rays is described. The method used to compute the exposure for a detector continuously growing with time is detailed. Systematic uncertainties in the exposure from the detector, the analysis and the involved physics are discussed. No tau neutrino candidates have been found. For neutrinos in the energy range 2×10^17 eV < E < 2×10^19 eV, assuming a diffuse spectrum of the form E^−2, data collected between 1 January 2004 and 30 April 2008 yield a 90% confidence-level upper limit of Eν^2dNντ/dEν<9×10−8 GeV cm−2 s−1 sr−1.