Destroy the Dark Brotherhood!

Background
This quest begins after completing Innocence Lost and starting With Friends Like These.... The Dragonborn is captured by Astrid and brought to an abandoned shack, where they are told to kill one of three captives inside. Killing Astrid instead will fail With Friends Like These... and begin this quest.

Report Astrid's death
After killing Astrid, the Dragonborn will be directed to report to any guard with the information that they have killed a Dark Brotherhood assassin. The guard will tell them to report to Commander Maro, who resides at the Penitus Oculatus Outpost in Dragon Bridge.

Maro says that he has been tracking the Dark Brotherhood and waiting for an opportunity to attack. On hearing news of the death of their leader, he tasks the Dragonborn with infiltrating the Dark Brotherhood Sanctuary using the password his agents recently acquired, and slaying every assassin inside.

Kill everyone in the Sanctuary!
Travel to the Sanctuary, near Falkreath. To complete the quest, the Dragonborn must kill Nazir, Gabriella, Arnbjorn, Festus Krex, Veezara, and Lis (the Frostbite Spider).

The assassins are somewhat powerful, and collectively possess a range of attacks that make fighting several of them in the cramped space fairly dangerous. Ironically, although they're assassins, it is particularly easy to sneak up on them. Use of bows, invisibility and poison are effective ways to take care of them.

Report back to Commander Maro
When the job is done, report back to Commander Maro to complete the quest and collect the reward of 3,000. Once the quest is complete, the Dark Brotherhood Sanctuary will remain accessible and undamaged, in contrast to finishing the Dark Brotherhood quest line when most of it is destroyed.

Trivia

 * Completing this quest is the fastest way to get the Blade of Woe, by looting it off Astrid. Looting Nazir is also the only way to obtain a red copy of a Redguard Hood.
 * Cicero and Babette are not killed during this quest.
 * 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 Dragonborn reports Astrid's death to a guard, they say "I killed the leader of the Dark Brotherhood," though there is no way in-game for the Dragonborn to know that she was their leader. If the Dragonborn survives an attack by a Dark Brotherhood Assassin, it will yield a note signed by Astrid, which gives some clue as to her position. Likewise, if the Dragonborn breaks into the room in the Black-Briar Manor basement where Maven Black-Briar keeps an effigy, there is a note To The Brotherhood addressed to Astrid, which further implies her leadership of the Brotherhood.
 * It is worth noting that completing the Dark Brotherhood questline plus all of the additional assassinations that Nazir has to offer yields 20,000, while destroying the Brotherhood earns only 3,000. Other quest rewards, such as Muiri's marriage option and Shadowmere, will likewise be forfeit.
 * The total amount of coinage that most players will achieve - with the included 20,000 coins, will range anywhere from around 29000 - 34000. This is including the side contracts that Nazir offers, and other missions such as Mourning Never Comes that are required to complete the questline for the Dark Brotherhood.
 * After killing Astrid, the three captives can be freed.

Restoring the Dark Brotherhood questline using console commands.
WARNING: USE AT OWN RISK
 * 1) Open console
 * 2) Type setstage db02a 10
 * 3) Go to the Abandoned Shack
 * 4) Enter the shack and find Astrid dead
 * 5) Open console
 * 6) Click Astrid and type resurrect
 * 7) Wait until she walks outside
 * 8) Follow her outside, and she will "fly" away
 * 9) Go to the Dark Brotherhood sanctuary
 * 10) Repeat step 5 and 6 on all the dead Brotherhood members