We report on the structure of a perpendicular MHD shock in the partially ionized gas of the local interstellar medium (LISM). In situations where the flow speed is sub-Alfvenic in the ion gas and supersonic in the atom gas, the ions and atoms behave differently in the shock. Ion-atom drifts provide a dissipation mechanism which permits computation of the steady state structure of the transition between upstream and downstream conditions. Using parameter ranges which are appropriate for the LISM, we find that the heliospheric bow shock may be many hundreds of AU thick. ¿ American Geophysical Union 1996