We report on the first complete computation of the mixed QCD-electroweak (EW) corrections to the neutral-current Drell-Yan process. Superseding previously applied approximations, our calculation provides the first result at this order that is valid in the entire range of dilepton invariant masses. The two-loop virtual contribution is computed by using semianalytical techniques, overcoming the technical problems in the evaluation of the relevant master integrals. The cancellation of soft and collinear singularities is achieved by a formulation of the q_{T} subtraction formalism valid in the presence of charged massive particles in the final state. We present numerical results for the fiducial cross section and selected kinematical distributions. At large values of the lepton p_{T} the mixed QCD-EW corrections are negative and increase in size, to about -15% with respect to the next-to-leading-order QCD result at p_{T}=500 GeV. Up to dilepton invariant masses of 1 TeV the computed corrections amount to about -1.5% with respect to the next-to-leading-order QCD result.