  Motivated by recent nuclear magnetic resonance experiments on
ZnCu$_3$(OH)$_6$Cl$_2$, we present an exact-diagonalization study of the
combined effects of non-magnetic impurities and Dzyaloshinskii-Moriya (DM)
interactions in the $s = 1/2$ kagome antiferromagnet. The local response to an
applied field and correlation-matrix data reveal that the dimer freezing which
occurs around each impurity for $D = 0$ persists at least up to $D/J\simeq
0.06$, where $J$ and $D$ denote respectively the exchange and DM interaction
energies. The phase transition to the ($Q = 0$) semiclassical, 120$^\circ$
state favored at large $D$ takes place at $D/J\simeq 0.1$. However, the dimers
next to the impurity sites remain strong up to values $D \sim J$, far above
this critical point, and thus do not participate fully in the ordered state. We
discuss the implications of our results for experiments on
ZnCu$_3$(OH)$_6$Cl$_2$.
