  We have carried out a detailed study of one of the most popular evolutionary
channels for the production of Type Ia supernova (SN Ia) progenitors, the
semi-degenerate channel (CO+MS), where a carbon/oxygen (CO) white dwarf (WD)
accretes matter from an unevolved or slightly evolved non-degenerate star until
it reaches the Chandrasekhar mass limit. Employing Eggleton's stellar evolution
code and adopting the prescription of Hachisu et al. (1999) for the accretion
efficiency, we have carried out full binary evolution calculations for about
2300 close WD binary systems and mapped out the initial parameters in the
orbital period -- secondary mass ($P$--$M_2$) plane (for a range of WD masses)
which lead to a successful Type Ia supernova. We obtained accurate, analytical
fitting formulae to describe this parameter range which can be used for binary
population synthesis (BPS) studies. The contours in the $P$--$M_2$ plane differ
from those obtained by Hachisu et al. (1999) for low-mass CO WDs, which are
more common than massive CO WDs. We show that white dwarfs with a mass as low
as $0.67M_\odot$ can accrete efficiently and reach the Chandrasekhar limit. We
have implemented these results in a BPS study to obtain the birthrates for SNe
Ia and the evolution of birthrates with time of SNe Ia for both a constant star
formation rate and a single star burst. The birthrates are somewhat lower than
(but comparable to) those inferred observationally.
