The Black Moon is never actually seen in-game. Although the presence of Black Moon Stones suggests its existence, it is unknown if it is even still in the skies or if it has somehow been destroyed. Its power of inversing gravity is implied in the Inverse Isle discovery, which also potentially suggests that the Black Moon could have been a victim of its own power, having been destroyed due to a black hole-like effect and the result was the creation of the Dark Rift.
The very existence of Black Moon Stones suggests that a Black Moon had to have existed at some point in the Old World. Furthermore, the fact that Old World Yafutoma was an active participant the Gigas Wars meant that there was not yet a Dark Rift separating them from the other civilizations. Otherwise, their only connection to the rest of the world was through the Green Continent, which would make it unlikely for them to want to participate in a global conflict. Therefore, if a Black Moon did exist, then it had to have existed before the Dark Rift came into being, and thus had to have existed as a whole in the Old World. It is interesting to note, however, that no one in the entire game, not even the Silvite elders, mention a Black Moon ever existing. As the the Silvite Elders had a highly detailed chronicle of the Old World, it is unlikely that they would not have known about a Black Moon had it actually existed. However, given how guarded they were with information, it is equally unlikely that they would have spoken of it freely. Though they seem comfortable with the fact that they were responsible for the death of millions through the Rains, it is still possible that they did not dare to disclose the existence of the Black Moon, even a devastating meteor shower would still pale in comparison to the atrocity of obliterating an entire moon. Then again, the Black Moon itself was never relevant to the current story, and thus did not need to be mentioned.
But that would also raise the question of why the Silvites would bother using Zelos, the Silver Gigas, on the Black Moon, assuming it even existed, when there is absolutely no evidence of a Black Gigas or even a Black Continent. One can assume that the Dark Rift is where a Black Continent used to be, as there is significant land mass within the Dark Rift, but then it would be impossible for a Black Civilization to NOT have interacted with the rest of the world, as it would be a center point between the old Yellow, Red, and Blue Civilizations.
The only shred of evidence that seems to directly hint at a '7th moon' is the Mysterious Rings discovery in the North Ocean, which can be interpreted as an astrological chart of the Arcadian planet system (1 planet, 7 moons in orbit). This chart seems to denote several interesting things: first and most relevant to this theory is that one of the orbital bodies being noted as having left the system by the lines around the drawing, second, the other 5 moons are linked by some common influence, while the final moon remains uninfluenced. This is perhaps an ancient recording of the fact that five moons were successfully subjected to the Rains of Destruction, barring the Silver Moon -- and further detailing the unanticipated effect that the Black Moon suffered -- namely it's removal from the system. Another possible interpretation is that the 7 circles surrounding the center circle is the seal of the Black Moon. Every other moon in Arcadia has a specific number associated with its seal, ranging from 1-6, so it's highly likely that the seal of 7 would belong to the hypothetical 7th moon.