Available: After the College Quest: Under Saarthal .
Go speak to Onmund and he tells you about a trade he did with Enthir. He wants you to go to him and ask him about the amulet. Locate and speak to Enthir. He is HARD to persuade and you can't buy the amulet from him either. Ask him if there's any way to get the amulet back.
Note: I WAS able to persuade him, but my speech skill was 75+ at the time, so it is possible to succeed at persuading him if your skill is high enough (if anyone knows the threshold here for the speech skill, let us know). Also, if you have the " Amulet of Articulation " from the Thieves guild, I would suspect it would help a TON.
You now need to get a Staff, the Staff of Charming , which is, as always, in a random location. You need to fast travel to the closest location, get the staff and return it to Enthir.
GLITCH NOTICE: If you ALREADY have the staff in your possession, the game will STILL tell you to "Go and get it" from somewhere (it told ME to go to Shriekwind Bastion). The fix to this is to equip it and go find a weapon rack somewhere and activate it (weapon racks are ALSO known to be glitchy, just FYI) so you put your staff on it (since you can't drop it). This will let you retrieve it.
Return to Enthir once you have the staff and he'll give you the amulet that Onmund want's so bad. Now simply go and give it back to Onmund, who will be overjoyed and will join you as a companion anytime you ask now.