The Imperial Library

The last piece, Ria Silmane visited her champion and said:

"I would take no chances, for Tharn has no mercy and will attack the moment he perceives the Staff reassembled. You must be swift. To the northeast lies the fire mountain, Dagoth-Ur. Somewhere in its volcanic depths is the final piece of the Staff of Chaos. The entrance to that fabled mountain has disappeared with the Dwarves that mined it. There may still be those who know of it, however. Find the entrance, enter the volcanic catacombs, and return with the final piece of the Staff. We are so close, I wish you well..."

Dagoth Ur is located in Morrowind province. Every Dark Elf knows the fiery mountain, however for the Eternal Champion, he had to do an errant for the ruler of Ebonheart first before he can get the location of the entrance of the mountain. The ruler of Ebonheart wanted the champion to recover the Hammer of Gharen in a dungeon known as the Black Gate. After finishing the errant, the champion was finally able to get into the dungeon of Dagoth Ur and recover the last piece.

"The sacred entrance of the Dwarven smiths into the fire mountain, Dagoth-Ur, stands before you.
Magical runes glow in the red firelight, shimmering with heat.
Sulphurous gouts of steam escape from vents within the rock face of the opening.
The final piece of the Staff of Chaos beckons..."

Jagar Tharn made another futile attempt to stop the Eternal Champion:

"I congratulate you on your success. Have you discovered however, what I have known all along? The Staff is drained of all magical potential. I did it myself before scattering the pieces across the realm. It is a useless stick, good for churning butter or rowing a boat, but not for saving the Emperor. Come find me if you dare, and I will relieve you of any further burdens you may think to face. My legions await..."

One more place to visit: The Imperial Palace.