  Observations have demonstrated that supernovae efficiently produce dust. This
is consistent with the hypothesis that supernovae and asymptotic giant branch
stars are the primary producers of dust in the Universe. However, there has
been a longstanding question of how much of the dust detected in the interiors
of young supernova remnants can escape into the interstellar medium. We present
new hydrodynamical calculations of the evolution of dust grains that were
formed in dense ejecta clumps within a Cas A-like remnant. We follow the
dynamics of the grains as they decouple from the gas after their clump is hit
by the reverse shock. They are subsequently subject to destruction by thermal
and kinetic sputtering as they traverse the remnant. Grains that are large
enough ($\sim 0.25\,\mu$m for silicates and $\sim 0.1\,\mu$m for carbonaceous
grains) escape into the interstellar medium while smaller grains get trapped
and destroyed. However, grains that reach the interstellar medium still have
high velocities, and are subject to further destruction as they are slowed
down. We find that for initial grain size distributions that include large
($\sim 0.25 - 0.5\,\mu$m) grains, 10--20\% of silicate grains can survive,
while 30--50\% of carbonaceous grains survive even when the initial size
distribution cuts off at smaller ($0.25\,\mu$m) sizes. For a 19 M$_{\odot}$
star similar to the progenitor of Cas A, up to 0.1 M$_{\odot}$ of dust can
survive if the dust grains formed are large. Thus we show that supernovae under
the right conditions can be significant sources of interstellar dust.
