Is it possible to hear a sonic boom when the aircraft is exactly at Mach one?
No, if standard atmospheric conditions apply.

Since speed of sound is proportional to temperature and temperature normally decreases with altitude, the speed of the aircraft at Mach 1 is subsonic in warmer, lower air. This means also that the Mach shock will diffuse and be audible on the ground either as a protracted rumble or not at all, depending on the aircraft's altitude.

Note that I assumed that the listener is located below the flight path of the aircraft. If that would not be the case, the noise of the aircraft diving below the listener altitude would be substantial.