  We present the star formation histories of 39 galaxies with high quality
rest-frame optical spectra at 0.5<z<1.3 selected to have strong Balmer
absorption lines and/or Balmer break, and compare to a sample of
spectroscopically selected quiescent galaxies at the same redshift. Photometric
selection identifies a majority of objects that have clear evidence for a
recent short-lived burst of star formation within the last 1.5 Gyr, i.e.
"post-starburst" galaxies, however we show that good quality continuum spectra
are required to obtain physical parameters such as burst mass fraction and
burst age. Dust attenuation appears to be the primary cause for
misidentification of post-starburst galaxies, leading to contamination in
spectroscopic samples where only the [OII] emission line is available, as well
as a small fraction of objects lost from photometric samples. The 31 confirmed
post-starburst galaxies have formed 40-90% of their stellar mass in the last
1-1.5 Gyr. We use the derived star formation histories to find that the
post-starburst galaxies are visible photometrically for 0.5-1 Gyr. This allows
us to update a previous analysis to suggest that 25-50% of the growth of the
red sequence at z~1 could be caused by a starburst followed by rapid quenching.
We use the inferred maximum historical star formation rates of several 100-1000
Msun/yr and updated visibility times to confirm that sub-mm galaxies are likely
progenitors of post-starburst galaxies. The short quenching timescales of
100-200 Myr are consistent with cosmological hydrodynamic models in which rapid
quenching is caused by the mechanical expulsion of gas due to an AGN.
