  The long-standing problem of existence of nowhere dense rational Julia set
with positive area has been solved by an example in quadratic polynomials by
Buff and Ch\'eritat. Since then many efforts have been devoted to finding out
new classes of rational maps with nowhere dense Julia sets having positive
area. So far, all known examples of this kind are renormalizable with only one
exception which is a quadratic polynomial. In this paper, by developing a new
approach, we prove that there exists a non-renormalizable cubic polynomial
having a Julia set with positive area.
