  We study how Type Ia/II supernova rates at various redshifts can be used to
constrain the cosmic star formation rate (CSFR) history. We use a
spectrophotometric model of galaxy evolution which provides a self-consistent
description of global properties of local galaxies. We have chosen two CSFR
histories, with and without dust extinction, both normalized in order to
reproduce the observed comoving luminosity densities at different wavelengths.
We show that as expected, the SNII rates provide a direct measurement of the
instantaneous CSFR, whereas the SNIa rate depends on the CSFR history during
the past few Gyrs. Therefore the SNIa rate detected up to $z {\sim}$ 1 is able
to provide new insight on the CSFR at higher redshift. The comparison with the
data are in good agreement with both local and high-$z$ measurement and
provides an additional evidence of the validity of the CSFR up to $z$ = 1, but
suggests values at $z >$ 1 higher than those of Madau et al. (1996) and
consistent with the presence of extinction. We also show that Type Ia supernova
observations at $z{\sim}$ 1 can already put constraints on the still puzzling
nature of Type Ia progenitors.
