  We have performed a numerical investigation of the ground state properties of
the frustrated quantum Heisenberg antiferromagnet on the square lattice
(``$J_1-J_2$ model''), using exact diagonalization of finite clusters with 16,
20, 32, and 36 sites. Using a finite-size scaling analysis we obtain results
for a number of physical properties: magnetic order parameters, ground state
energy, and magnetic susceptibility (at $q=0$). For the unfrustrated case these
results agree with series expansions and quantum Monte Carlo calculations to
within a percent or better. In order to assess the reliability of our
calculations, we also investigate regions of parameter space with
well-established magnetic order, in particular the non-frustrated case $J_2<0$.
We find that in many cases, in particular for the intermediate region $0.3 <
J_2/J_1 < 0.7$, the 16 site cluster shows anomalous finite size effects.
Omitting this cluster from the analysis, our principal result is that there is
N\'eel type order for $J_2/J_1 < 0.34$ and collinear magnetic order (wavevector
$\bbox{Q}=(0,\pi)$) for $J_2/J_1 > 0.68$. There thus is a region in parameter
space without any form of magnetic order. Including the 16 site cluster, or
analyzing the independently calculated magnetic susceptibility we arrive at the
same conclusion, but with modified values for the range of existence of the
nonmagnetic region. We also find numerical values for the spin-wave velocity
and the spin stiffness. The spin-wave velocity remains finite at the
magnetic-nonmagnetic transition, as expected from the nonlinear sigma
