  We present dense-gas--tracing molecular observations of six resolved Giant
Molecular Clouds (GMCs) in the Andromeda Galaxy (M31). Using the NOEMA
interferometer, we observed the transitions of HCN(1-0), HCO$^+$(1-0), and
HNC(1-0), as well as $^{13}$CO(1-0) and 100 GHz continuum emission. This
complements our earlier work with the Submillimeter Array (SMA), including
resolved dust continuum detections of these clouds at 230 GHz. In this work, we
first compare different continuum measurements to conclude that the average
free-free contamination of the observed flux is 71% at 3 mm but only 13% at 1
mm, confirming that emission at 3 mm is less reliable than that at 1 mm for
calculating dust masses of star-forming clouds. While the $^{13}$CO emission is
more extended than both HCN and HCO$^+$ emission, which in turn is more
extended than HNC emission, we find that both HCN and HCO$^+$ are spatially
coincident with, and similarly extended as, the 230 GHz dust emission. This
suggests that both the 230 GHz dust continuum and most importantly the HCN
emission traces the dense gas component of these GMCs. From comparison of the
molecular emission with dust masses derived from the 230 GHz continuum
emission, we obtain the first direct measurements of the dust-mass-to-light
ratios ($\alpha^\prime_{HCN}$ and $\alpha^\prime_{HCO^+}$) in GMCs of an
external galaxy. For HCN, the result is broadly similar to a measurement in the
local Perseus cloud suggesting that these are indeed dense gas conversion
factors. A larger cloud sample will be required to assess whether HCN is
tracing comparable cloud-scale density regimes across the environments of M31.
