  Today, Type Ia supernovae are essential tools for cosmology, and recognized
as major contributors to the chemical evolution of galaxies. The construction
of detailed supernova progenitor models, however, was so far prevented by
various physical and numerical difficulties in simulating binary systems with
an accreting white dwarf component, e.g., unstable helium shell burning which
may cause significant expansion and mass loss. Here, we present the first
binary evolution calculation which models both stellar components and the
binary interaction simultaneously, and where the white dwarf mass grows up to
the Chandrasekhar limit by mass accretion. Our model starts with a 1.6 Msun
helium star and a 1.0 Msun CO white dwarf in a 0.124 day orbit. Thermally
unstable mass transfer starts when the CO core of the helium star reaches 0.53
Msun, with mass transfer rates of 1...8 times 10^{-6} Msun/yr. The white dwarf
burns the accreted helium steadily until the white dwarf mass has reached ~ 1.3
Msun and weak thermal pulses follow until carbon ignites in the center when the
white dwarf reaches 1.37 Msun. Although the supernova production rate through
this channel is not well known, and this channel can not be the only one as its
progenitor life time is rather short (~ 10^7 - 10^8 yr), our results indicate
that helium star plus white dwarf systems form a reliable route for producing
Type Ia supernovae.
