Under certain conditions, the free surface of a conducting liquid subject to an electric field elongates into a cone whose apex emits a thin stationary jet that carries an electric current. The structure of the flow in the cone-to-jet transition region is investigated here, assuming that the size of this region is small compared with any other length of the system where the conical meniscus is formed. The local problem depends then on three non-dimensional parameters, two of which are properties of the liquid while the third measures the flow rate injected through the meniscus. Numerical solutions are computed and the electric current is determined as a function of these parameters. A qualitative asymptotic analysis of the physically important limit of large non-dimensional flow rates gives an electric current increasing as the square root of the flow rate and independent of the dielectric constant of the liquid. When the inertia of the liquid is taken into account, the flow in this asymptotic limit is effectively inviscid in the bulk of the transition region, where the electric current is dominated by conduction in the liquid and the surface is close to an equipotential of the electric field in the gas. The effects of the viscosity of the liquid, the current transported by convection of the surface charge, and the electric shear at the surface come into play in a slender region of the jet. The limit of small non-dimensional flow rates is briefly discussed.