  The present paper is a continuation of our previous work on the stochastic
quantization of the $\exp(\Phi)_2$-quantum field model on the two-dimensional
torus. Making use of key properties of Gaussian multiplicative chaos and
refining the method for singular SPDEs introduced in the previous work, we
construct a unique time-global solution to the corresponding parabolic
stochastic quantization equation in the full "$L^{1}$-regime"
$\vert\alpha\vert<\sqrt{8\pi}$ of the charge parameter $\alpha$. We also
identify the solution with an infinite-dimensional diffusion process
constructed by the Dirichlet form approach.
