We present a new \@element [ ] [ 12 ] [ ] [ ] { \mathrm { CO } } ( J=1–0 ) –line survey of the Andromeda galaxy , M 31 , with the highest resolution to date ( 23 \arcsec , or 85 pc along the major axis ) , observed On-the-Fly with the IRAM 30-m telescope . We mapped an area of about 2 \degr \times 0 \aas@@fstack { \circ } 5 which was tightly sampled on a grid of 9 \arcsec 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 _ { \mathrm { mb } } -scale . Emission from the \@element [ ] [ 12 ] [ ] [ ] { \mathrm { CO } } ( 1–0 ) line is detected from galactocentric radius R = 3 kpc to R = 16 kpc , but peaks in intensity at R \sim 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 i 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 i at an angular resolution of 45 \arcsec . In several selected regions we also observed the \@element [ ] [ 12 ] [ ] [ ] { \mathrm { 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 i , FIR at 175 \mu m and radio continuum , and interpret the CO data in terms of molecular gas column densities . For a constant conversion factor X _ { \mathrm { 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 \simeq 10 kpc . We also compare the distributions of H i , H _ { 2 } and total gas with that of the cold ( 16 K ) dust traced at \lambda 175 \mu m. The ratios N ( { H \textsc { i } } ) / I _ { 175 } and ( N ( { H \textsc { i } } ) +2 N ( \mathrm { H } _ { 2 } ) ) / I _ { 175 } increase by a factor of \sim 20 between the centre and R \simeq 14 kpc , whereas the ratio 2 N ( \mathrm { H } _ { 2 } ) / I _ { 175 } only increases by a factor of 4 . For a constant value of X _ { \mathrm { CO } } , this means that either the atomic and total gas–to–dust ratios increase by a factor of \sim 20 or that the dust becomes colder towards larger radii . A strong variation of X _ { \mathrm { 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 ( \mbox { H } _ { 2 } ) = 3.6 \times 10 ^ { 8 } \mbox { M } _ { \sun } at the adopted distance of 780 kpc . This is 12 % of the total neutral gas mass within this radius and 7 % of the total neutral gas mass in M 31 .