The Imperial Library

After the third piece is liberated, Ria Silmane visited the champion again in his dream:

"You have done well. I had thought you lost to me. The third piece of the Staff is yours. Rumors have been whispered about in the Empire. I think Tharn senses that the pieces come together. You must take care to not be discovered by him. Your death would be most unpleasant. I have divined the location of the fourth piece of the Staff. It lies in the Halls of Colossus, a structure built to honor a race of Giants, though none have been seen in the land for ages. It is said that the Halls of Colossus sat perched on a cliff overlooking the Great Divide, which is along the southern edge of Tamriel. Perhaps the Halls are near that coastline. I only wish I could offer you more. Fare thee well my friend, and find this place, for the sake of our loved ones..."

The place is located in Elsweyr province. To get the place, the Eternal Champion had to consult with Mages Guild in Corinth. Some days before, the warrior-priests of Temple of Agamanus attacked the Guild, and stole a stone tablet. It held the key to deciphering a part of the Elder Scrolls, the part that shows the location of the Halls of Colossus. And after the trouble with the warrior-priests of Temple of Agamanus, the champion was able to deliver the tablet, and in turn was given the exact location of the Halls of Colossus.

"You stand before the gigantic walls of the Halls of Colossus, built by giants in an age long past.
The wind echoes mournfully through the empty battlements.
Here lies the fourth piece of the Staff of Chaos."

As usual Jagar Tharn showed up after the Eternal Champion acquired the piece:

"It is obvious, whoever you are, that you are a cunning and worthy foe. I will not insult your intelligence with further digression. My minions soon approach, but I will not consign you to the dark door of Death. You have proven your worth. Once you are killed, I will raise you to fight for me, as one servants. Give in to me now and I offer you everlasting life..."

As the previous encounters, the Eternal Champion outwitted Tharn's minions.