  We classify the minimum volume smooth complex hyperbolic surfaces that admit
smooth toroidal compactifications, and we explicitly construct their
compactifications. There are five such surfaces and they are all arithmetic,
i.e., they are associated with quotients of the ball by an arithmetic lattice.
Moreover, the associated lattices are all commensurable. The first
compactification, originally discovered by Hirzebruch, is the blowup of an
Abelian surface at one point. The others are bielliptic surfaces blown up at
one point. These appear to be the first known examples of smooth toroidal
compactifications birational to bielliptic surfaces.
