Destroy the Dark Brotherhood!

Walkthrough
Before starting this quest, one must complete the quest Innocence Lost.

The quest begins after meeting Astrid, the Highest Ranking Member of the Dark Brotherhood in Skyrim. When you get to the Abandoned Shack, you meet Astrid and she gives you a choice of 3 captives to kill, one of which is apparently the target of a contract. This quest triggers if you instead fight and kill her. Warning: she is difficult to kill, as she is a Dark Brotherhood assassin. Killing Astrid will also get you the Blade of Woe much faster than the Dark Brotherhood questline.

While harming Astrid, the quest With Friends Like These... will fail. After her death, this quest starts.

Report Astrid's death to a guard
In the first objective of the quest, you are told to report to any guard with the information that you have killed a Dark Brotherhood assassin.

Dragonborn: "I killed the leader of the Dark Brotherhood."

City Guard: "You did what? By the gods, you're serious. You'd better report this to Commander Maro right away. He's at the Penitus Oculatus Outpost, up in Dragon Bridge."

Speak with Commander Maro
Travel to Dragon Bridge and speak to Maro in the Outpost.

Dragonborn: "A guard sent me. I killed the leader of the Dark Brotherhood."

Maro: "The leader of the Dark Brotherhood? You mean Astrid? Dead? And this is no jest? You, my friend. You've slain their leader. This honor should be yours. Do this, and you will be rewarded most handsomely!"

Kill everyone in the Sanctuary!
Maro commands you to travel to the Dark Brotherhood Sanctuary, in which the rest of the assassins reside, and kill all the assassins. They are somewhat powerful, and what's suggested is that you kill them off one by one, by way of sneak. Ironically, although they're assassins, it is particularly easy to sneak up on them. You get a fair amount of expensive gear from the assassins.

Report back to Commander Maro
When you report back, Commander Maro gives you a reward of 3,000 Septims. Once the quest is complete, the Dark Brotherhood Sanctuary will remain accessible and undamaged at all times, in contrast to finishing the Dark Brotherhood quest line when most of it is destroyed. After the completion of this quest, all Dark Brotherhood quests will not be accessible.

Trivia

 * There is a Stone of Barenziah on Astrid's dresser in the Sanctuary
 * The player does not kill Cicero or Babette in this quest. Since their fates and the fate of the Night Mother are undecided, it is possible that in future Elder Scrolls games the Dark Brotherhood can rebuild itself even after the Dovahkiin wiped out the last Dark Brotherhood Sanctuary.
 * After killing Nazir, be sure to loot his body, as he is wearing the only red copy of a Redguard Hood (however invaluable).
 * Guards can be heard saying afterwards, "I could've gone into that sanctuary and killed all those Dark Brotherhood types but I was... sick that day."
 * When the player reports Astrid's death to a guard, they report "I killed the leader of the Dark Brotherhood", but there is no way in-game for the Dovahkiin to know that she was specifically their leader. However if the Dovahkiin survives an attempted assassination by a Dark Brotherhood Assassin, it will yield a note signed by Astrid giving some clue to her position.
 * After reporting in to Commander Maro he will rant about how he will tell the emperor himself of what you accomplished.
 * After Telling Commander Maro The Dark Brotherhood was destroyed, and is "ranting" you can repeatedly keep hitting him for weapon style experience (one & two handed weapons, bows, and magicka) you just have to wait for him to get up.
 * It is worth knowing that you get a lot more money by completing the Dark Brotherhood questline plus all of the additional assassinations that Nazir has to offer. (3,000 compared to 20,000)
 * The player should also know that if they choose to destroy the Dark Brotherhood, they cannot marry Muiri.