The Andromeda galaxy (M 31) hosts one of the nearest and most quiescent super-massive black holes, which provides a rare, but promising opportunity for studying the physics of black hole accretion at the lowest state. We have conducted a multi-frequency, multi-epoch observing campaign, using the Karl G. Jansky Very Large Array (VLA) in its extended configurations in 2011-2012, to advance our knowledge of the still poorly known radio properties of M 31*. For the first time, we detect M 31* at 10, 15 and 20 GHz, and measure its spectral index, α ≈ −0.45 ± 0.08 (S ν ∝ ν α ), over the frequency range of 5-20 GHz. The relatively steep spectrum suggests that the observed radio flux is dominated by the optically-thin part of a putative jet, which is located at no more than a few thousand Schwarzschild radii from the black hole. On the other hand, our sensitive radio images show little evidence for an extended component, perhaps except for several parsec-scale "plumes", the nature of which remains unclear. Our data also reveal significant (a few tens of percent) flux variation of M 31* at 6 GHz, on timescales of hours to days. Furthermore, a curious decrease of the mean flux density, by ∼50%, is found between VLA observations taken during 2002-2005 and our new observations, which might be associated with a substantial increase in the mean X-ray flux of M 31* starting in 2006.