  The influence of dilution with non--magnetic impurities on the order in the
ground state of the $S=1/2$ Heisenberg antiferromagnet is investigated by
Quantum Monte Carlo simulations. Data of the spin correlation functions and
thermodynamic properties for system sizes up to $L\times L=16\times16$ and for
impurity concentrations up to $\delta=37.5\%$ are presented. In the low doping
regime the correlation length shows similiar dependence on temperature as in
the pure case. The staggered magnetization $\langle N_z^2(\delta)\rangle$ in
the ground state which is the order parameter of the present model is estimated
by extrapolating data at low temperatures. Impurity concentration dependence of
$\langle N_z^2(\delta)\rangle$ is presented. $\langle N_z^2(\delta)\rangle$
decreases monotonically with impurity concentration and becomes very small
around $\delta_c \sim 0.35$ which is much smaller than the classical
percolation threshold.
