  A stationary Poisson line tessellation is considered whose directional
distribution is concentrated on two different atoms with some positive weights.
The shape of the typical cell of such a tessellation is studied when its area
or its perimeter tends to zero. In contrast to known results where the area or
the perimeter tends to infinity, it is shown that the asymptotic shape of cells
having small area is degenerate. Again in contrast to the case of large cells,
the asymptotic shape of cells with small perimeter is not uniquely determined.
The results are accompanied by a large scale simulation study.
