  The low-temperature coarsening dynamics of a one-dimensional Ising model,
with conserved magnetisation and subject to a small external driving force, is
studied analytically in the limit where the volume fraction \mu of the minority
phase is small, and numerically for general \mu. The mean domain size L(t)
grows as t^{1/2} in all cases, and the domain-size distribution for domains of
one sign is very well described by the form P_l(l) \propto
(l/L^3)\exp[-\lambda(\mu)(l^2/L^2)], which is exact for small \mu (and possibly
for all \mu). The persistence exponent for the minority phase has the value 3/2
for \mu \to 0.
