A properly embedded holomorphic disc in the ball with finite area and dense boundary curve

In this paper we construct a properly embedded holomorphic disc in the unit ball ${\mathbb{B}}^2$ of ${\mathbb{C}}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on ${\mathbb{B}}^2$; on the other hand, its boundary curve is everywhere dense in the sphere $b{\mathbb{B}}^2$.