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I have two corals sharing a small rock - two types of tree coral. One has grown much faster than the other and has crowded the other. The smaller one appears to slowly be dying/shrinking/shriveling. I'd like to move it to it's own rock before I lose it.

What's entailed in removing a coral from one rock and securing it to another rock?

I would just break the rock instead of trying to remove the coral (possibly damaging it) and trying to get it to attach to a new rock.
live rock is pretty easy to break.