  We investigate the quantum phase transitions in the half-filled Hubbard model
on the triangular lattice by means of the path-integral renormalization group
(PIRG) method with a new iteration scheme proposed recently. It is found that
as the Hubbard interaction $U$ increases, the paramagnetic metallic state
undergoes a first-order phase transition to a nonmagnetic insulating (NMI)
state at $U_{c1}\sim 7.7t$, which is followed by another first-order transition
to a $120^\circ$ N\'eel ordered state at $U_{c2}\sim 10t$, where $t$ is the
transfer integral. Our results ensure the existence of the intermediate NMI
phase, and resolve some controversial arguments on the nature of the previously
proposed quantum phase transitions. We find that
$\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ is located in the NMI phase close to the
metal-insulator transition point.
