  We report magnetic and thermodynamic properties of a $4d^1$ (Mo$^{5+}$)
magnetic insulator MoOPO$_4$ single crystal, which realizes a $J_1$-$J_2$
Heisenberg spin-$1/2$ model on a stacked square lattice. The specific-heat
measurements show a magnetic transition at 16 K which is also confirmed by
magnetic susceptibility, ESR, and neutron diffraction measurements. Magnetic
entropy deduced from the specific heat corresponds to a two-level degree of
freedom per Mo$^{5+}$ ion, and the effective moment from the susceptibility
corresponds to the spin-only value. Using {\it ab initio} quantum chemistry
calculations we demonstrate that the Mo$^{5+}$ ion hosts a purely spin-$1/2$
magnetic moment, indicating negligible effects of spin-orbit interaction. The
quenched orbital moments originate from the large displacement of Mo ions
inside the MoO$_6$ octahedra along the apical direction. The ground state is
shown by neutron diffraction to support a collinear N\'eel-type magnetic order,
and a spin-flop transition is observed around an applied magnetic field of 3.5
T. The magnetic phase diagram is reproduced by a mean-field calculation
assuming a small easy-axis anisotropy in the exchange interactions. Our results
suggest $4d$ molybdates as an alternative playground to search for model
quantum magnets.
