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Data collected at the Pierre Auger Observatory are used to establish an upperlimit on the diffuse flux of tau neutrinos in the cosmic radiation.Earth-skimming $\nu_{\tau}$ may interact in the Earth's crust and produce a$\tau$ lepton by means of charged-current interactions. The $\tau$ lepton mayemerge from the Earth and decay in the atmosphere to produce a nearlyhorizontal shower with a typical signature, a persistent electromagneticcomponent even at very large atmospheric depths. The search procedure to selectevents induced by $\tau$ decays against the background of normal showersinduced by cosmic rays is described. The method used to compute the exposurefor a detector continuously growing with time is detailed. Systematicuncertainties in the exposure from the detector, the analysis and the involvedphysics are discussed. No $\tau$ neutrino candidates have been found. Forneutrinos in the energy range $2\times10^{17}$ eV $< E_{\nu}$ $<$$2\times10^{19}$ eV, assuming a diffuse spectrum of the form $E_{\nu}^{-2}$,data collected between 1 January 2004 and 30 April 2008 yield a 90%confidence-level upper limit of $E_\nu^{2}\mathrm{d}N_{\nu_\tau}/\mathrm{d}E_{\nu} < 9 \times 10^{-8}$ GeV cm$^{-2}$s$^{-1}$ sr$^{-1}$.

Limit on the diffuse flux of ultra-high energy tau neutrinos with the surface detector of the Pierre Auger Observatory