  It is now well established that Zn doping of high-$T_C$ cuprates reduces
their $T_C$ and triggers the appearence of a spin glass phase. In this context,
we have solved exactly the problem of N non magnetic impurities in the
staggered flux phase of the Heisenberg model which we assume to be a good
mean-field approximation for the spin-gap phase of the cuprates. In this model,
the quasiparticule spectrum has four nodes on the Fermi surface, and
linearization of the spectrum in the neighbourhood of these nodes leads to a
system of 2D Dirac fermions. In the presence of a macroscopic number of (non
magnetic) impurities, the problem has a characteristic logarithmic structure
that renders ineffective the usual perturbative expansions. We have used this
logarithmic structure to calculate an exact solution. For a concentration ni of
impurities in the unitary scattering limit, the additional density of states is
found to be proportional to $ni/(w \ln^2 (|w|/D))$ (where D is the infrared
cut-off of the linearized spectrum) in analogy with the 1D case of doped
spin-Peierls and two-leg ladders compounds. We argue that the system exhibits a
quasi long-range order at T=0 with instantaneous spin-spin correlations
decreasing as $ni/ \ln^4 (ni/R_{ij})$ for large distances $R_{ij}$ between
impurity sites. We predict enhanced low energy fluctuations and compare these
results to NMR and inelastic neutron scattering experiments in the high-$T_C$
cuprates.
