Yes, you can get a new one. I've done the quest twice and I've got two cubes.
You have to redo parts of the quests to forge a new one, but you don't have to do EVERYTHING over again.
The parts that you need to do again are:
* Obtaining Sphere
* Wrecked Opticorder Forge
* Kill the Last Planegazer again to loot The Cube (Mission: Dolis - The Last Planegazer)
* The Missing Piece of the Cube
Now just go back to Gray Island and reforge your cube with those three components by using them on the Opticording Combinator and then using the machine itself.