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In this paper we present new observations of the interstellar medium in the non-interacting edge-on galaxy NGC 5907. We have observed the J=2-1 and J=1-0 lines of the ¹²CO molecule and radio continuum emission at λ1.2mm. The distribution of the molecular gas (as traced by CO) shows a maximum in the central region and a ring or spiral arm at r∼7kpc. Further analysis of the major axis distribution reveals evidence for an inner ring-like structure at r∼3.5kpc. The kinematics can be described by rigid rotation in the inner part, a turnover at ∼3kpc, and differential rotation with a velocity of 230km/s in the outer disk. The observed continuum emission is mainly due to thermal radiation of cold dust with an average temperature of T_d=18K, with a small gradient from 20K to 16K from the centre to the outer disk. This cold dust component is necessary to explain our results. The dust emission closely follows the molecular gas in the central region, but is also detected at large radii where no CO can be seen. In these regions the dust absorption cross section per H atom at λ1.2mm is estimated to be σ_λ^HI∼4.5x10^–27cm², a value similar to that in the outer parts of other galaxies. From the λ1.2mm emission we estimated a molecular mass of NGC 5907 of 0.9x10⁹M_☉, about 50% smaller than from the CO emission. By combining the CO and continuum data we found that the CO-H₂-conversion ratio increases with galactocentric radius, from ∼0.7x10²⁰ at the centre to ∼1.6x10²⁰cm^–2/K.km/s at r=7.5kpc. A comparison of NGC 5907 and other edge-on galaxies concerning gas distribution, central kinematics and dust properties is presented.