  This article is based on the methods developed in [AGG]. We construct a
complex hyperbolic structure on a trivial disc bundle over a closed orientable
surface $\Sigma$ (of genus 2) thus solving a long standing problem in complex
hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers
also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface
of genus >1 admits a holomorphically fillable contact structure. The
constructed example M satisfies the relation $2(\chi+e)=3\tau$ which is
necessary for the existence of a holomorphic section of the bundle, where
$\chi=\chi\Sigma$ stands for the Euler characteristic of $\Sigma$, e=eM, for
the Euler number of the bundle, and $\tau$, for the Toledo invariant. (The
relation is also valid for the series of examples constructed in [AGG].) Open
question: Does there exist a holomorphic section of the bundle M?
