  We make a new proposal to describe the very low temperature susceptibility of
the doped Haldane gap compound Y$_2$BaNi$_{1-x}$Zn$_x$O$_5$. We propose a new
mean field model relevant for this compound. The ground state of this mean
field model is unconventional because antiferromagnetism coexists with random
dimers. We present new susceptibility experiments at very low temperature. We
obtain a Curie-Weiss susceptibility $\chi(T) \sim C / (\Theta+T)$ as expected
for antiferromagnetic correlations but we do not obtain a direct signature of
antiferromagnetic long range order. We explain how to obtain the ``impurity''
susceptibility $\chi_{imp}(T)$ by subtracting the Haldane gap contribution to
the total susceptibility. In the temperature range [1 K, 300 K] the
experimental data are well fitted by $T \chi_{imp}(T) = C_{imp} (1 + T_{imp}/T
)^{-\gamma}$. In the temperature range [100 mK, 1 K] the experimental data are
well fitted by $T \chi_{imp}(T) = A \ln{(T/T_c)}$, where $T_c$ increases with
$x$. This fit suggests the existence of a finite N\'eel temperature which is
however too small to be probed directly in our experiments. We also obtain a
maximum in the temperature dependence of the ac-susceptibility $\chi'(T)$ which
suggests the existence of antiferromagnetic correlations at very low
temperature.
