Abstract. We present a new 12 CO(J=1-0)-line survey of the Andromeda galaxy, M 31, with the highest resolution to date (23 ′′ , or 85 pc along the major axis), observed On-the-Fly with the IRAM 30-m telescope. We mapped an area of about 2 • × 0.• 5 which was tightly sampled on a grid of 9′′ with a velocity resolution of 2.6 km s −1 . The r.m.s. noise in the velocity-integrated map is around 0.35 K km s −1 on the T mb -scale. Emission from the 12 CO(1-0) line is detected from galactocentric radius R = 3 kpc to R = 16 kpc, but peaks in intensity at R ∼ 10 kpc. Some clouds are visible beyond R = 16 kpc, the farthest of them at R = 19.4 kpc. The molecular gas traced by the (1-0) line is concentrated in narrow arm-like filaments, which often coincide with the dark dust lanes visible at optical wavelengths. The H arms are broader and smoother than the molecular arms. Between R = 4 kpc and R = 12 kpc the brightest CO filaments and the darkest dust lanes define a two-armed spiral pattern that is well described by two logarithmic spirals with a constant pitch angle of 7• -8• . Except for some bridge-like structures between the arms, the interarm regions and the central bulge are free of emission at our sensitivity. The arm-interarm brightness ratio averaged over a length of 15 kpc along the western arms reaches about 20 compared to 4 for H at an angular resolution of 45 ′′ . In several selected regions we also observed the 12 CO(2-1)-line on a finer grid. Towards the bright CO emission in our survey we find normal ratios of the (2-1)-to-(1-0) line intensities which are consistent with optically thick lines and thermal excitation of CO. We compare the (velocity-integrated) intensity distribution of CO with those of H , FIR at 175 µm and radio continuum, and interpret the CO data in terms of molecular gas column densities. For a constant conversion factor X CO , the molecular fraction of the neutral gas is enhanced in the spiral arms and decreases radially from 0.6 on the inner arms to 0.3 on the arms at R ≃ 10 kpc. We also compare the distributions of H , H 2 and total gas with that of the cold (16 K) dust traced at λ175 µm. The ratios N(H i)/I 175 and (N(H i) + 2N(H 2 ))/I 175 increase by a factor of ∼ 20 between the centre and R ≃ 14 kpc, whereas the ratio 2N(H 2 )/I 175 only increases by a factor of 4. For a constant value of X CO , this means that either the atomic and total gas-to-dust ratios increase by a factor of ∼ 20 or that the dust becomes colder towards larger radii. A strong variation of X CO with radius seems unlikely. The observed gradients affect the cross-correlations between gas and dust. In the radial range R = 8-14 kpc total gas and cold dust are well correlated; molecular gas is better correlated with cold dust than atomic gas. At smaller radii no significant correlations between gas and dust are found. The mass of the molecular gas in M 31 within a radius of 18 kpc is M(H 2 ) = 3.6 × 10 8 M ⊙ at the adopted distance of 780 kpc. This is 12% of the total neutral gas mass within this radius and 7%...