The surface detector array of the Pierre Auger Observatory is sensitive to Earth-skimming tau neutrinos that interact in Earth’s crust. Tau leptons from ντ charged-current interactions can emerge and decay in the atmosphere to produce a nearly horizontal shower with a significant electromagnetic component. The data collected between 1 January 2004 and 31 August 2007 are used to place an upper limit on the diffuse flux of ντ at EeV energies. Assuming an E⁻²ν differential energy spectrum the limit set at 90% C.L. is E2νdNντ/dEν<1.3×10⁻⁷  GeV cm⁻² s⁻¹ sr⁻¹ in the energy range 2×10¹⁷ eV<Eν<2×10¹⁹ eV.