We investigate the formation of ringlike deposits in drying drops of DNA. In analogy with the colloidal "coffee rings," DNA is transported to the perimeter by the capillary flow. At the droplet edge, however, DNA forms a lyotropic liquid crystal (LC) with concentric chain orientations to minimize the LC elastic energy. During the final stages of drying, the contact line retracts, and the radial stress causes undulations at the rim that propagate inward through the LC and form a periodic zigzag structure. We examine the phenomenon in terms of a simple model based on LC elasticity.