We present a detailed study of how the Star Formation Rate (SFR) relates to the interstellar medium (ISM) of M31 at ~140pc scales. The SFR is calculated using the far-ultraviolet and 24um emission, corrected for the old stellar population in M31. We find a global value for the SFR of 0.25+/-0.05Msun/yr and compare this with the SFR found using the total far-infrared (FIR) luminosity. There is general agreement in regions where young stars dominate the dust heating. Atomic hydrogen (HI) and molecular gas (traced by carbon monoxide, CO) or the dust mass is used to trace the total gas in the ISM. We show that the global surface densities of SFR and gas mass place M31 amongst a set of low-SFR galaxies in the plot of Kennicutt (1998b). The relationship between SFR and gas surface density is tested in six radial annuli across M31, assuming a power law relationship with index, N. The star formation law using total gas traced by HI and CO gives a global index of N=2.03+/-0.04, with a significant variation with radius; the highest values are observed in the 10kpc ring. We suggest that this slope is due to HI turning molecular at ~10Msun/pc2. When looking at H2 regions, we measure a higher mean SFR suggesting a better spatial correlation between H2 and SF. We find N~0.6 with consistent results throughout the disk - this is at the low end of values found in previous work and argues against a superlinear SF law on small scales.
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