Scanning tunneling microscopy and electrical conductivity of redox molecules in conducting media (aqueous or other media) acquire increasing importance both as novel single-molecule science and with a view on molecular scale functional elements. Such configurations require full and independent electrochemical potential control of both electrodes involved. We provide here a general formalism for the electric current through a redox group in an electrochemical tunnel contact. The formalism applies broadly in the limits of both weak and strong coupling of the redox group with the enclosing metal electrodes. Simple approximate expressions better suited for experimental data analysis are also derived. Particular attention is given to the effects of the Debye screening of the electric potential in the narrow tunneling gap based on the limit of the linearized Poisson-Boltzmann equation. The current/overpotential relation shows a maximum at a position which depends on the ionic strength. It is shown, in particular, that the dependence of the maximum position on the bias voltage may be nonmonotonous. Approximate expressions for the limiting value of the slope of the current/overpotential dependence and the width of the maximum on the bias voltage are also given and found to depend strongly on both the Debye screening and the position of the redox group in the tunnel gap, with diagnostic value in experimental data analysis.