Abstract: We will show that, for any value $ c \leq 1$ of the central charge, there exists a consistent CFT which is diagonal and has a continuous spectrum. This CFT is based on the structure constants found by Schomerus (2003), Zamolodchikov(2005) and Petkova and Kostov(2005). The interpretation of these structure constants as a part of a consistent CFT has been dismissed until Viti and Delfino (2011) showed that they have a natural application in critical statistical models. We will show that this theory differs from the generelised minimal models, as well as from other theories at rational values of the central charge such as the minimal models or the Runkel-Watts kind of theories. We will argue that the solution we found extends the definition and consistency of the Liouville theory to all complex values of c. Our claims are backed by numerical tests of crossing symmetry. Finally, we will discuss the application of such theory to describe certain many-point observables in critical statistical models.