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Background
After becoming a vampire (by contacting Sanguinare Vampiris from a vampire and leaving it untreated for three days), if you ask innkeepers about rumors, then they'll tell you that Falion in Morthal studies the undead. This information will trigger the quest. If you have the Dawnguard DLC, and if you become a vampire while working with the Dawnguard, then Isran will order you to get cured by Falion in Morthal, and this order can trigger the quest as well.
Walkthrough
When you talk to Falion inside his house in Morthal, he'll tell you that he can perform a ritual to cure you of vampirism, but only if you bring him a filled
Black Soul Gem. Conveniently, Falion will sell you empty
Black Soul Gems, so you'll just need to purchase one (if necessary) and then use it to capture the soul of a sentient creature (such as a forsworn or falmer).
When you bring the filled
Black Soul Gem to Falion, he'll ask you to meet him at dawn (between 5:00 am and 7:00 am) at the summoning circle to the north of town. When you do so, he'll perform the ritual as promised, and you'll return to normal. You can complete this quest as many times as necessary.