We present a continuum theory for ribbons composed of two thin elastic rods, here termed strands, that are bound together by elastic forces.  The ribbon is modeled as a special Cosserat macro-rod endowed with microstructure parameters that give the relative positions and orientations of the strands with respect to the position and  orientation of the macro-rod. We derive constitutive relations for the ribbon and discuss possible applications of this theory to DNA molecules.