  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 $b\mathbb{B}^2$.
