  We study the spin-$1\over2$ Heisenberg antiferromagnet with an
antiferromagnetic $J_3$ (third nearest neighbor) interaction on a square
lattice. We numerically diagonalize this ``$J_1$-$J_3$'' model on clusters up
to 32-sites and search for novel ground state properties as the frustration
parameter $J_3/J_1$ changes. For ``larger'' $J_3/J_1$ we find enhancement of
incommensurate spin order, in agreement with spin-wave, large-$N$ expansions,
and other predictions. But for intermediate $J_3/J_1$, the low lying excitation
energy spectrum suggests that this incommensurate order is short-range. In the
same region, the first excited state has the symmetries of the columnar dimer
(spin-Peierls) state. The columnar dimer order parameter suggests the presence
of long-range columnar dimer order. Hence, this spin-Peierls state is the best
candidate for the ground state of the $J_1$-$J_3$ model in an intermediate
$J_3/J_1$ region.
