Adamantite ore is obtained through the Mining skill in various places throughout SoulSplit by mining Adamantite rocks. It can be smelted through the Smithing skill to form an Adamant bar, which can then be smithed into various types of adamant weapons and armor.
Adamant rocks can be found at Neitiznot North (1) and Falador Cave (3) (There are other places, but these are the easiest to reach with the available teleports)