  We demonstrate that a nonzero concentration $n_v$ of static, randomly-placed
vacancies in graphene leads to a density $w$ of zero-energy quasiparticle
states at the band-center $\epsilon=0$ within a tight-binding description with
nearest-neighbour hopping $t$ on the honeycomb lattice. We show that $w$
remains generically nonzero in the compensated case (exactly equal number of
vacancies on the two sublattices) even in the presence of hopping disorder, and
depends sensitively on $n_v$ and correlations between vacancy positions. For
low, {\em but not-too-low} $|\epsilon|/t$ in this compensated case, we show
that the density of states (DOS) $\rho(\epsilon)$ exhibits a strong divergence
of the form $\rho_{\rm 1D}(\epsilon) \sim |\epsilon|^{-1}/
[\log(t/|\epsilon|)]^{(y+1)} $, which crosses over to the universal low-energy
asymptotic form expected on symmetry grounds $\rho_{\rm GW}(\epsilon) \sim
|\epsilon|^{-1}e^{-b[\log(t/|\epsilon|)]^{2/3} }$ below a crossover scale
$\epsilon_c \ll t$. $\epsilon_c$ is found to decrease rapidly with decreasing
$n_v$, while $y$ decreases much more slowly.
