We study ion condensation on a patterned surface with stripes of alternating charge. The competition between adsorbed ion-ion and adsorbed ion-surface interactions leads to the formation of different strongly correlated structures of condensed ions in the low-temperature limit (LTL). We consider two types of arrangements which have lowest energy in the LTL: (1) ions adsorbed onto the stripe center lines and (2) arrays of dipoles at the interfaces between charged domains. We determine the preferred arrangement as a function of surface charge density, the chemical potential of the ions in the surrounding medium, and the geometric parameters of the system. We determine the conditions for the appearance of more complex ionic patterns by considering simple perturbations of the stripe-centered and dipolar array structures.