  The ground state of the square lattice bilayer quantum antiferromagnet with
nearest ($J_1$) and next-nearest ($J_2$) neighbour intralayer interaction is
studied by means of the dimer expansion method up to the 6-th order in the
interlayer exchange coupling $J_3$. The phase boundary between the spin-gap
phase and the magnetically ordered phase is determined from the poles of the
biased Pad\'e approximants for the susceptibility and the inverse energy gap
assuming the universality class of the 3-dimensional classical Heisenberg
model. For weak frustration, the critical interlayer coupling decreases
linearly with $\alpha (= J_2/J_1)$. The spin-gap phase persists down to $J_3=0$
(single layer limit) for $0.45 \simleq \alpha \simleq 0.65$. The crossover of
the short range order within the disordered phase is also discussed.
