  We investigate the dispersion in the internal colours of faint spheroidals in
the HDFs North and South. We find that a remarkably large fraction ~30% of the
morphologically classified spheroidals with I<24 mag show strong variations in
internal colour, which we take as evidence for recent episodes of
star-formation. In most cases these colour variations manifest themselves via
the presence of blue cores, an effect of opposite sign to that expected from
metallicity gradients. Examining similarly-selected ellipticals in five rich
clusters with 0.37<z<0.83 we find a significant lower dispersion in their
internal colours. This suggests that the colour inhomogeneities have a strong
environmental dependence being weakest in dense environments where spheroidal
formation was presumably accelerated at early times. We use the trends defined
by the cluster sample to define an empirical model based on a high-redshift of
formation and estimate that at z~1 about half the field spheroidals must be
undergoing recent episodes of star-formation. Using spectral synthesis models,
we construct the time dependence of the density of star-formation. Although the
samples are currently small, we find evidence for an increase in $\rho_{SFR}$
between z=0 to z=1. We discuss the implications of this rise in the context of
that observed in the similar rise in the abundance of galaxies with irregular
morphology. Regardless of whether there is a connection our results provide
strong evidence for the continued formation of field spheroidals over 0<z<1.
