  In this paper, I investigate the possible quantization, in the context of
LQG, of three dimensional gravity in the case of positive cosmological constant
{\Lambda} and try to make contact with alternative quantization approaches
already existing in the literature. Due to the appearance of an anomaly in the
constraints algebra, previously studied as a first step of the analysis,
alternative techniques developed for the quantization of systems with
constraints algebras not associated with a structure Lie group need to be
adopted. Therefore, I introduce an ansatz for a physical state which gives some
transition amplitudes in agreement with what one would expect from the
Turaev-Viro model. Moreover, in order to check that this state implements the
right dynamicss, I show that it annihilates the master constraint for the
theory up to the first order in {\Lambda}.
