  Doubly-degenerate binary systems consisting of two white dwarfs both composed
of carbon and oxygen and close enough that mass is transferred from the less
massive to the more massive are possible progenitors of type Ia supernovae. If
the mass transfer rate is slow enough that the accreting white dwarf can reach
1.38 solar masses then it can ignite carbon degenerately at its centre. This
can lead to a thermonuclear runaway and thence a supernova explosion. However
if the accretion rate is too high the outer layers of the white dwarf heat up
too much and carbon ignites there non-degenerately. A series of mild carbon
flashes can then propagate inwards and convert the carbon to neon relatively
gently. There is no thermonuclear runaway and no supernova. We have determined
the critical rate for accretion and find it to be generally about two-fifths of
the Eddington rate. In the case of actual bianry stars the mass transfer rate
falls off as mass is transferred so we find that this rate can be very much
exceeded at the onset of mass transfer. We determine combinations of white
dwarf masses that can lead to central degenerate carbon ignition.
