We discuss recent observations of CO, CS, and NH, emission lines from two molecular clouds in the inner regions of M31. The intensities of the (CO)-C-12(1-0), (CO)-C-12(2-1), (CO)-C-13(1-0), and (CO)-C-13(2-1) lines have been measured with the IRAM 30 m telescope at the same angular resolution of 23''. Upper limits have been obtained in the CS(2-1), CS(3-2) and in the NH3(1, 1) and (2, 2) lines (the latter two with the Effelsberg 100 m telescope). The line intensities and ratios are compared with the predictions of a one-side UV-illuminated plane-parallel model which treats consistently radiative, chemical, and excitation precesses of CO and its isotopes. A satisfactory agreement cannot be achieved for a single-density model. It appears that the emission of the (CO)-C-12 lines is dominated by a low-density, very cold gas while that of the (CO)-C-13 lines comes largely from higher density clumps inside the clouds. The properties of the molecular clouds can be understood as resulting mainly from a very small rate of photodissociation due to a very low UV radiation field, together with a low cosmic-ray density. Typical kinematic temperatures inside the clouds can drop to values less than 5 K.