  We present spectra of 1142 colour-selected stars in the direction of the
Sagittarius Dwarf Spheroidal (Sgr dSph) galaxy, of which 1058 were taken with
VLT/FLAMES multi-object spectrograph and 84 were taken with the SAAO Radcliffe
1.9-m telescope grating spectrograph. Spectroscopic membership is confirmed (at
>99% confidence) for 592 stars on the basis of their radial velocity, and
spectral types are given. Very slow rotation is marginally detected around the
galaxy's major axis. We identify five S stars and 23 carbon stars, of which all
but four carbon stars are newly-determined and all but one (PQ Sgr) are likely
Sgr dSph members. We examine the onset of carbon-richness in this metal-poor
galaxy in the context of stellar models. We compare the stellar death rate (one
star per 1000-1700 years) to known planetary nebula dynamical ages and find
that the bulk population produce the observed (carbon-rich) planetary nebulae.
We compute average lifetimes of S and carbon stars as 60-250 and 130-500 kyr,
compared to a total thermal-pulsing asymptotic giant branch lifetime of
530-1330 kyr. We conclude by discussing the return of carbon-rich material to
the ISM.
