A model for the transport of electrons in a semiconductor is
considered where the electrons travel with saturation speed
in the direction of the driving force computed self consistently
from the Poisson equation. Since the velocity is discontinuous
at zeroes of the driving force, an interpretation of the model
in the distributional sense is not necessarily possible. For a
spatially one-dimensional model existence of distributional
solutions is shown by passing to the limit in a regularized
problem corresponding to a scaled drift-diusion model with a
velocity saturation assumption on the mobility. Several explicit
solutions of the limiting problem are computed and compared to
the results of numerical computations.