  Persistence is considered in diffusion--limited cluster--cluster aggregation,
in one dimension and when the diffusion coefficient of a cluster depends on its
size $s$ as $D(s) \sim s^\gamma$. The empty and filled site persistences are
defined as the probabilities, that a site has been either empty or covered by a
cluster all the time whereas the cluster persistence gives the probability of a
cluster to remain intact. The filled site one is nonuniversal. The empty site
and cluster persistences are found to be universal, as supported by analytical
arguments and simulations. The empty site case decays algebraically with the
exponent $\theta_E = 2/(2 - \gamma)$. The cluster persistence is related to the
small $s$ behavior of the cluster size distribution and behaves also
algebraically for $0 \le \gamma < 2$ while for $\gamma < 0$ the behavior is
stretched exponential. In the scaling limit $t \to \infty$ and $K(t) \to
\infty$ with $t/K(t)$ fixed the distribution of intervals of size $k$ between
persistent regions scales as $n(k;t) = K^{-2} f(k/K)$, where $K(t) \sim
t^\theta$ is the average interval size and $f(y) = e^{-y}$. For finite $t$ the
scaling is poor for $k \ll t^z$, due to the insufficient separation of the two
length scales: the distances between clusters, $t^z$, and that between
persistent regions, $t^\theta$. For the size distribution of persistent regions
the time and size dependences separate, the latter being independent of the
diffusion exponent $\gamma$ but depending on the initial cluster size
distribution.
