  We theoretically study the subband structure of single Si $\delta$-doped GaAs
inserted in a quantum well and subject to an electric field applied along the
growth direction. We use an efficient self-consistent procedure to solve
simultaneously the Schr\"odinger and Poisson equations for different values of
electric field and temperature. We thus find the confining potential, the
subband energies and their corresponding envelope functions, the subband
occupations, and the oscillator strength of intersubband transitions. Opposite
to what is usually the case when dealing with the quantum-confined Stark effect
in ordinary quantum wells, we observe an abrupt drop of the energy levels
whenever the external field reaches a certain value. This critical value of the
field is seen to depend only slightly on temperature. The rapid change in the
energy levels is accompanied by the appearance of a secondary well in the
confining potential and a strong decrease of the oscillator strength between
the two lowest subbands. These results open the possibility to design devices
for use as optical filters controlled by an applied electric field.
