Morrowind:Mephala's Quest

Quick Quest
#Find the Altar to Mephala in the Vivec Morag Tong Guildhall.
#Talk to Taros Dral about sensitive matters.
#Travel to Balmora and enter Balyn Omarel's House unseen.
#Place the Treated Bittergreen Petals in the Balyn's Cauldron.
#Return to Taros Dral at the Altar to Mephala.

Detailed Walkthrough

Altar to Mephala
The Altar to Mephala is not in a likely setting for a Daedric Shrine. It is hidden in the secret Morag Tong Guildhall in Vivec's Arena Canton, specifically the Storage Area of the Canalworks. It is also the only Daedra shrine that does not feature a statue of a Daedra Lord, rather one that is similar to an Imperial Cult altar.  

Sensitive Matters
Speak to Taros Dral about sensitive matters. He will not say much up front, just something about Mephala giving him information which must be acted upon. If you inquire further, you'll learn that Mephala wishes you to poison Balyn Omavel, an important member of the Morag Tong who has been performing assassinations without writs. Dral will supply Treated Bittergreen Petals to put in Balyn's Cauldron.

Balyn Omavel's House
  can be found in Balmora, south of the eastern guard tower. Balyn is wandering around the front, and if he sees you, the quest will be ruined. In order to sneak into his house unseen, use the trapdoor found on the roof of the house. Once inside, interact with the cauldron on the west wall. Answer "yes" when asked "Would you like to place the poison in the pot?" Escape the same way you entered, making sure not to be spotted by Balyn.

Reward
Return to Taros Dral and he'll tell you to speak to the altar. Do so and you will receive the Ring of Khajiit as a reward. If you return to Balyn Omavel's house you'll find him dead on the floor.

Notes
Do not take anything in Balyn Omavel's house before completing this quest. Doing so will crash the game when you talk to Mephala as the game will try to disable something that does not exist.
When looking at the doors to his house, it incorrectly displays his last name as "Omarel". The Morrowind Patch Project fixes this discrepancy by changing the NPC name and all quest dialogue to "Balyn Omarel", which is easier to do than changing the name of a cell.
The key to this quest is speed more than stealth. If you are less than 1024 units away from Omavel and less than 257 units away from the door for more than five seconds while doing the quest, he will spot you and ruin the reward. As long as you take less than five real time seconds to enter his house, you can do it in full view. To give some rough idea of required positions: 257 units from the door equates to the middle of the landing outside. In that position, 1024 units away is approximately between the two buttresses on the northern wall of Hecerinde's house. If Omavel is wandering close to the River Odai, you will be safe.
An easy way to do this quest is to enter Balyn's house before you get the quest. Pick the lock unseen, enter Omavel's house, cast "Mark", exit, get the quest, cast "Recall".
While this quest involves the Morag Tong, and starts within their headquarters, you do not actually need to be a member. It's certainly easier to do the quest as a member, as joining the guild gives you a free Disposition boost with Taros Dral, but a high Personality or a little Persuasion will work as well.
A glitch may occur that involves you finding Balyn dead inside his home before you have poisoned him. This should not prevent the quest's completion; just poison the pot as usual and report back to Taros Dral.
After becoming Grandmaster of the Morag Tong, Dral may cease speaking about "sensitive matters", making the quest impossible to activate. On the PC, the following console commands can be used to start the quest and gain the necessary items:
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: 
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You could also just add the topic by entering:
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It is important to note that Omavel is a member of the Morag Tong. If you have already joined the Morag Tong before doing this quest, getting caught breaking into his home may get you expelled, even if Omavel himself does not catch you.
If you kill Omavel by some method other than using the poison, Taros Dral's disposition towards you will drop by 20 points next time you talk to him.
Quest stage 20 is never set during the quest.
It is possible to get 10 samples of Treated Bittergreen Petals from Dral. This is done by pickpocketing 5 of them from him before speaking to him about "sensitive matters". He will give you another 5 once you agree to help him.

Quest Stages