We report on a first-principles study of the effects produced by a static electric field on proton conduction in ordinary hexagonal ice (phase Ih) and in its proton-ordered counterpart (phase XI). We performed ab initio molecular dynamics simulations of both phases and investigated the effects produced by the field on the structure of the material, with particular attention paid to the phenomenon of proton transfer. We observed that in ice Ih molecules start to dissociate for field intensities around 0.25 V/Å, as in liquid water, whereas fields stronger than 0.36 V/Å are needed to induce a permanent proton flow. In contrast, in ice XI, electric fields as intense as 0.22 V/Å are already able to induce and sustain, through correlated proton jumps, an ionic current; this behavior suggests, somewhat counterintuitively, that the ordering of protons favors the autoprotolysis phenomenon. However, the same is not true for static conductivities. In fact, both crystalline phases show an ohmic behavior in the conduction regime, but the conductivity of ice Ih turns out to be larger than that of ice XI. We finally discuss the qualitative and quantitative importance of the conspicuous concentration of ionic defects generated by intense electric fields in determining the value of the conductivity, also through a comparison with the experimental data available for saline ices.