  We have determined the ground state for both a ladder array of Josephson
junctions and a ladder of thin superconducting wires. We find that the
repulsive interaction between vortices falls off exponentially with separation.
The fact that the interaction is short-range leads to novel phenomena. The
ground state vortex density exhibits a complete devil's staircase as the
applied magnetic field is increased, each step producing a pair of
metal-insulator transitions. The critical fields in the staircase are all
calculated analytically and depend only on the connectivity of the ladder and
the area of the elementary plaquette. In particular the normal square ladder
contains no vortices at all until the flux per plaquette reaches 0.5/sqrt{3}
flux quanta.
