  We develop the non-perturbative reduced phase space quantization of causal
diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant.
In Part I we described the classical reduction process and the reduced phase
space, $\widetilde{\mathcal P} = T^*(\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb
R))$, while in Part II we discuss the quantization of the phase space and
quantum aspects of the causal diamonds. Because the phase space does not have a
natural linear structure, a generalization of the standard canonical
(coordinate) quantization is required. In particular, as the configuration
space is a homogeneous space for the $\text{Diff}^+\!(S^1)$ group, we apply
Isham's group-theoretic quantization scheme. We propose a quantization based on
(projective) unitary irreducible representations of the $\text{BMS}_3$ group,
which is obtained from a natural prescription for extending
$\text{Diff}^+\!(S^1)$ into a transitive group of symplectic symmetries of the
phase space. We find a class of suitable quantum theories labelled by a choice
of a coadjoint orbit of the Virasoro group and an irreducible unitary
representation of the corresponding little group. The most natural choice,
justified by a Casimir matching principle, corresponds to a Hilbert space
realized by wavefunctions on $\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R)$
valued in some unitary irreducible representation of $\text{SL}(2, \mathbb R)$.
A surprising result is that the twist of the diamond corner loop is quantized
in terms of the ratio of the Planck length to the corner perimeter.
