Maybe the elves conjured it by condensing magical essence into incredibly hard, but light-weight, bands similar to metal which were woven into armor.
or, maybe the dwarves mine both adamantite and mithril, but no self-respecting dwarf would use mithril as armor. Only the weakest would need to lighten the armor like that, and it didn't offer as much protection as adamantite, anyways.
The elves on the other hand love the stuff, and buy most of it from the dwarves. In exchange, the elves provide magical enchantments superior to anything the dwarves could manage.
The dwarves and elves still really don't like each other (much the same way I really hate pushy salespeople), but economics would be a factor in any decision either side made in going to war.
However, if they did go to war, I bet the elves have a war surplus in mithril to help hold them out through the war and the dwarves have more than one customer when it comes to manufactured goods.
Also, it would be an excellent oppurtunity for smugglers when 2 sides economically dependent on each other went to war.
Hmm, that sparked a new idea for a game.