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. For neutrinos in the energy range 2x10(17) eV { E-nu { 2x10(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)dN(nu tau)/dE(nu){ 9x10(-8) GeV cm(-2) s(-1) sr(-1).