I know you have to craft or buy double-melded weapons in order to proceed with the quest. My question is: Do you get to keep these weapons?

For instance, in order to get the Gae Bolg, you need to get an Obelisk with double melded Savage Might IV materia. Do you get to keep this weapon afterward, or do you give it up in exchange for the relic?

Because I want Gae Bolg for my Dragoon, but I realize it's DRG only, so if I'm just playing LNC that double melded Obelisk could definitely come in handy...