  We present here estimates of the average rates of accretion of neutral gas
onto main-sequence galaxies and the conversion of atomic gas to molecular gas
in these galaxies at two key epochs in galaxy evolution: (i) $z\approx1.3-1.0$,
towards the end of the epoch of peak star-formation activity in the Universe,
and (ii) $z\approx1-0$, when the star-formation activity declines by an order
of magnitude. We determine the net gas accretion rate $\rm{R_{Acc}}$ and the
molecular gas formation rate $\rm{R_{Mol}}$ by combining the relations between
the stellar mass and the atomic gas mass, the molecular gas mass, and the
star-formation rate (SFR) at three epochs, $z=1.3$, $z=1.0$, and $z=0$, with
the assumption that galaxies evolve continuously on the star-forming
main-sequence. We find that, for all galaxies, $\rm{R_{Acc}}$ is far lower than
the average SFR $\rm{R_{SFR}}$ at $z\approx1.3-1.0$; however, $\rm{R_{Mol}}$ is
similar to $\rm{R_{SFR}}$ during this interval. Conversely, both $\rm{R_{Mol}}$
and $\rm{R_{Acc}}$ are significantly lower than $\rm{R_{SFR}}$ over the later
interval, $z\approx1-0$. We find that massive main-sequence galaxies had
already acquired most of their present-day baryonic mass by $z\approx1.3$. At
$z\approx1.3-1.0$, the rapid conversion of the existing atomic gas to molecular
gas was sufficient to maintain a high average SFR, despite the low net gas
accretion rate. However, at later times, the combination of the lower net gas
accretion rate and the lower molecular gas formation rate leads to a decline in
the fuel available for star-formation, and results in the observed decrease in
the SFR density of the Universe over the last 8 Gyr.
