After you show Momodi the letter from Owyne, she asks if you are willing to assist Ul'dah, in a matter of utmost secrecy.
This quest is only available if you started the game in Ul'dah, and you have completed the quest Secrets and Lies.
Starting the Quest
Talk to Momodi in the Adventurers' Guild in Ul'dah to start the quest. There will be a cutscene.
Deliver the Letter
The next quest destination is near the Alchemists' Guild in Ul'dah. Talk to the Sultansworn Elite and say Yes to proceed into the Heart of the Sworn. Then go forward and talk to Owyne. Hand over the letter, and there will be a cutscene.
Rendezvous with Owyne
The next destination is in Central Thanalan. Go east from Black Brush Station to find the destination. There will be a glow around it, indicating another instanced battle. Check on it and there will be a cutscene, and then the battle will ensue.
Solo Duty: Duty, Honor, Country
First, kill the Lawless Bowmen who are attacking Owyne, then kill the Lawless Fistfighters. When more enemies appear, kill the Lawless Bowmen attacking Owyne, then kill the Lawless Cursedealer and Lawless Spellweaver. Afterward, attack the Lawless Fistfighter, but when a Blanga appears, focus your attacks on it until it dies. Then kill the Fistfighter if he is still alive, and finally, focus your attacks on Garibald the Fargone until the battle ends. There will be a cutscene.
Afterward, another battle will will begin. Attack the Lesser Gargoyle and ignore the Masked Mage for now. After a while, Thancred will arrive and assist you in battle. When the Lesser Gargoyle dies, focus your attacks on the Masked Mage until he dies.
There will be another cutscene. This one is important, so you might want to pay attention. You can go to an inn room and use The Unending Journey to watch it again if you wish.
Speak with Papashan
Go back to Ul'dah and go to the Alchemists' Guild, and talk to the Sultansworn Elite to enter the Heart of the Sworn. Talk to Papashan. There will be a cutscene.
Next Main Scenario Quest
After completing the quest, talk to Owyne to accept the next quest: A Matter of Tradition.