  Magnetic properties of AMoOPO$_4$Cl (A = K, Rb) with Mo$^{5+}$ ions in the
4$d^1$ electronic configuration are investigated by magnetization, heat
capacity and NMR measurements on single crystals, combined with powder neutron
diffraction experiments. The magnetization measurements reveal that they are
good model compounds for the spin-1/2 $J_1-J_2$ square lattice magnet with the
first and second nearest-neighbor interactions. Magnetic transitions are
observed at around 6 and 8 K in the K and Rb compounds, respectively. In
contrast to the normal N\'eel-type antiferromagnetic order, the NMR and neutron
diffraction experiments find a columnar antiferromagnetic order for each
compound, which is stabilized by a dominant antiferromagnetic $J_2$. Both
compounds realize the unusual case of two interpenetrating $J_2$ square
lattices weakly coupled to each other by $J_1$.
