  We investigate a new experimental possibility of measuring the Newtonian
gravitational constant $G$ by using the weak measurement. Amplification via
weak measurement is one of the interesting phenomena of quantum mechanics. In
this letter, we consider it in a system consisting of many cold atoms which are
gravitationally interacting with an external macroscopic source and show that
it is possible to obtain ${\mathcal{O}}(10^{3})$ amplification of their
relative motion compared with the classical motion when the number of atoms are
${\mathcal{O}}(10^{15})$ and the observing time is $\sim 0.5$s. This result
suggests that it might be possible to use this system as a new experimental set
up for determining $G$. Besides, our study indicates that the gravitational
force can behave as a repulsive force because of the weak measurement.
