  We study C/O white dwarfs with masses of 1.0 to 1.4 Msun accreting
solar-composition material at very high accretion rates. We address the secular
changes in the WDs, and in particular, the question whether accretion and the
thermonuclear runaways result is net accretion or erosion. The present
calculation is unique in that it follows a large number of cycles, thus
revealing the secular evolution of the WD system.
  We find that counter to previous studies, accretion does not give rise to
steady state burning. Instead, it produces cyclic thermonuclear runaways of two
types. During most of the evolution, many small cycles of hydrogen ignition and
burning build a helium layer over the surface of the white dwarf. This He layer
gradually thickens and progressively becomes more degenerate. Once a sufficient
amount of He has accumulated, several very large helium burning flashes take
place and expel the accreted envelope, leaving no net mass accumulation.
  The results imply that such a system will not undergo an accretion induced
collapse, nor will it lead to a SN Type Ia, unless a major new physical process
is found.
