  Can a white dwarf, accreting hydrogen-rich matter from a non-degenerate
companion star, ever exceed the Chandrasekhar mass and explode as a type Ia
supernova? We explore the range of accretion rates that allow a white dwarf
(WD) to secularly grow in mass, and derive limits on the accretion rate and on
the initial mass that will allow it to reach $1.4M_\odot$ --- the Chandrasekhar
mass. We follow the evolution through a long series of hydrogen flashes, during
which a thick helium shell accumulates. This determines the effective helium
mass accretion rate for long-term, self-consistent evolutionary runs with
helium flashes. We find that net mass accumulation always occurs despite helium
flashes. Although the amount of mass lost during the first few helium shell
flashes is a significant fraction of that accumulated prior to the flash, that
fraction decreases with repeated helium shell flashes. Eventually no mass is
ejected at all during subsequent flashes. This unexpected result occurs because
of continual heating of the WD interior by the helium shell flashes near its
surface. The effect of heating is to lower the electron degeneracy throughout
the WD, and especially in the outer layers. This key result yields helium
burning that is quasi-steady state, instead of explosive. We thus find a
remarkably large parameter space within which long-term, self-consistent
simulations show that a WD can grow in mass and reach the Chandrasekhar limit,
despite its helium flashes.
