Jauffre has been told by Captain Burd of the Bruma guards that an Oblivion Gate has opened up just outside of town and he asks that you come along, seeing as you have experience with closing Oblivion Gates. After all, you are the Hero of Kvatch.
You must go just outside of Bruma near the east gate . There you meet Captain Burd who will ask you for a minute with his men. After giving a speech, he will ask you if you are ready, then charge into the Gate. There will be a few daedra on the way to the Gate, so pick them off first. Try and keep Burd and the guards alive for the extra damage they do to the daedra. Burd is a plot character, so he will only be knocked unconscious, but the guards can die. You can replenish their health by waiting an hour when the area is clear. If you keep waiting when they are damaged, it is highly possible that none of the guards will die. It is recommended that you try to keep them alive for the extra damage they do.
Once inside the Oblivion Gate, start following the path around counterclockwise. When you are as far north as you can go, begin heading south and west toward the center of the island and the citadel. The entrance is on the southwest side. Beware of the spinning defense turrets that hit you with fireballs when you're nearby. Enter the Fury Spike and then take the exit to the Rending Halls. Follow the path up and then take the east exit up again until you're as high as you can go, then take the central exit back to the Fury Spike. The exit one level below also leads to the Fury Spike, but is a dead end. Ascend the side walls and exit into the Corridors of Dark Salvation.
Once there, take either path and follow it up, perform a u-turn and then up again to the upper level which has another exit to the Fury Spike (the place with the lava flowing upward). Again take the side ramps up until you find the locked (but manageable, even without the Sigil Keeper nearby) doors to Sigillum Sanguis. Once inside, you should know the drill. Defeat the last line of daedra defense in the circular chamber, head up the stairs, defeat more daedra, up some stairs and then take the Sigil Stone, which transports you back outside of Bruma. Captain Burd will thank you and say that he will be able to close any other Oblivion Gates that pop up. Now it's time to go and see Jauffre and Martin for your next quests.
Alternatively, if you are more of the sneaking kind, you can make your way up to the sigil stone alone and proceed to grab it. The game will tell you that you should have done this with Captain Burd, so make your way back down to the bloodshrine and wait for Captain Burd to arrive with 2 guards completely unharmed.
Upon receiving the quest
- Martin told me that Jauffre needs to speak with me.
After speaking with Jauffre:
- Jauffre told me that an Oblivion Gate has opened near Bruma, and the Countess has asked for the Blades to help her guardsmen close it. I should find Captain Burd outside the city walls and help him close the Gate.
After speaking with Captain Burd:
- Captain Burd and his men are ready to help me close the Oblivion Gate. I just need to let Burd know when I'm ready.
After telling Burd you're ready:
- Captain Burd and his men are heading into the Oblivion Gate with me. I need to accompany Burd to the Sigil Stone chamber of the Oblivion Gate and show him how to close it.
After reaching the Sigil Stone:
- Burd and I have reached the Sigil Stone chamber. Now to close the Gate.
After closing the Oblivion Gate:
- I closed the Gate with Captain Burd. I should ask him if he needs any more help from me.
After speaking to Captain Burd:
- Captain Burd thanked me for my help, and said that he and his guardsmen will do their best to close any new Oblivion Gates that open near Bruma. I should report my success to Jauffre.