Circle graphs are derived from the intersections of a set of chords. They have applications in VLSI design, bioinformatics, and chemistry. Some intractable problems on general graphs can be solved in polynomial time on circle graphs. As such, the study of circle graph algorithms has received attention. State-of-the-art algorithms for finding the maximum weight clique of a circle graph are very efficient when the graph is sparse. However, these algorithms require Θ(n2) time when the graph is dense. We present an algorithm that is a factor of √n faster for dense graphs in which many chord endpoints are shared. We also argue that these assumptions are practical.