We have performed joint ASCA + ROSAT PSPC spectral fitting on the inner 5' of the bulge of M31. We find that single-component spectral models provide an inadequate fit to the spectrum, in contrast to previous studies by Einstein and Ginga. Although the 2-10 keV spectrum can be fitted adequately with a bremsstrahlung model with kT = 7.4 +/- 0.3 keV, an additional soft component with kT = 0.38 +/- 0.03 keV is required to fit the spectrum below 2 keV. This soft component comprises 38% +/- 6% of the total emission in the 0.1-2.0 keV band, and possibly more depending on the absorption value used in the lit. Since previous spatial studies of the bulge of M31 indicate that less than 25% of the X-ray emission from the bulge in this band is from a diffuse gaseous component, this implies that stellar sources, namely, low-mass X-ray binaries (LMXBs), are responsible for some if not all of the soft component. The spectral properties of M31 are very similar to those of the X-ray-faint early-type galaxy NGC 4382. This supports the claim that the unexplained soft X-ray emission seen previously in these galaxies also emanates from LMXBs.