[Zigûr]

Interestingly the online encyclopedias seem to like claiming that Balin's expedition recovered Durin's Axe, but all Gandalf says is "I can read no more for a long way... except the word gold, and Durin's Axe and something helm."

For all we know, the book said "We were unable to find Durin's Axe," or as Inzil has suggested, "Durin's Axe" may not have even referred to a particular object. I think people like the idea of Balin swinging this impressive heirloom, but it seems to me like it can't be read decisively either way. Perhaps it is possible that the "helm" referred to might be Durin's actual crown (as opposed to the one made of stars) but that's altogether conjectural. I suppose if the Dwarves did find Durin's Axe, it's possible that either the orcs took it back or it was buried with Balin.

It's an interesting inclusion given that while we hear a good deal about Dwarf-made weapons of note, especially those forged by Telchar, I believe this is one of the only if not the only instance of what might be a specifically identified weapon used by the Dwarves themselves (as opposed to, say, Gimli's axe). One imagines it might have been an heirloom of Durin's line. In any event, unlike the Last of the Seven, it did not escape Moria with Thráin I. If there was such an axe, presumably Náin I had it with him when he was slain by the Balrog.