  The evolution of binary systems consisting of an asymptotic giant branch star
of mass equal to 3 M_sun or 5 M_sun, and a main sequence star of mass equal to
0.4 M_sun or 0.6 M_sun with orbital periods > 200 days has been followed from
the onset through the late stages of the common envelope phase. Using a nested
grid technique, the three-dimensional hydrodynamical simulations of an
asymptotic giant branch star with radii approximately 1 A.U. indicate that a
significant fraction of the envelope gas is unbound (about 31% and 23% for
binaries of 3 M_sun and 0.4 M_sun, and 5 M_sun and 0.6 M_sun respectively) by
the ends of the simulations, and that the efficiency of the mass ejection
process is about 40%. While the original volume of the giant is virtually
evacuated in the late stages, most of the envelope gas remains marginally bound
on the grid. At the ends of our simulations, when the orbital decay timescale
exceeds about 5 years, the giant core and companion orbit one another with a
period of about 1 day (2.4 days for a binary involving a more evolved giant),
although this is an upper limit to the final orbital period. For a binary of 5
M_sun and 0.4 M_sun, the common envelope may not be completely ejected.
