  We present a numerical study of the spin-1/2 bilayer Heisenberg
antiferromagnet with random interlayer dimer dilution. From the temperature
dependence of the uniform susceptibility and a scaling analysis of the spin
correlation length we deduce the ground state phase diagram as a function of
nonmagnetic impurity concentration p and bilayer coupling g. At the site
percolation threshold, there exists a multicritical point at small but nonzero
bilayer coupling g_m = 0.15(3). The magnetic properties of the single-layer
material La_2Cu_{1-p}(Zn,Mg)_pO_4 near the percolation threshold appear to be
controlled by the proximity to this new quantum critical point.
