An unresolved radio source with a flux density of 28 +/- 5-mu-Jy has been detected at the center of the galaxy M31. This source coincides with the optical stellar nucleus and with a variable Einstein X-ray source but is only 1/30 as luminous as the Galactic center source Sgr A and one-fifth as luminous as Sgr A*. Consequently, this object is the weakest known nuclear radio source. No evidence is found for radio emission from the remnant of supernova 1885A in M31. An upper limit of 15-mu-Jy (3-sigma) has been determined for the flux density of this source, which is 1/15 the value for Tycho's SNR if it were at the same distance.