SOLUTIONS: I had the same problem with the XBOX 360 Oblivion Clavicus Vile / Umbra turn in. I saved right before
I talked to the Daedric statue again to complete the quest. I decided to keep the sword (you can give it back and take the mask if you want-- mask adds +20 to personality, and is really ugly, especially for a girly like me). The game would always freeze about 15 seconds after talking to Clavicus Vile no matter what I did. I searched the Internet for hours, running back and forth to the game, trying different methods of getting around this problem. Only ONE method worked for me and I'm so happy I could cry!! After talking to the statue and chosing to KEEP the sword, I fast traveled to EAST CHEYDINHAL GATE!! It worked! The dog talked to me, and the dog statuette was removed from my inventory.
Here's some methods that worked for other people, but didn't work for me:
*After speaking to statue (Clavicus Vile), hit your 'back' button on controller, and "wait" 3 hours. Do this 6 consecutive times. Then fast travel to Water District in Imperial City. If you have never used your "wait" feature, test it out a few times and get familiar with it, it's a very handy tool!!
*Fast travel anywhere over and over for a few minutes until the dog statue removes from your inventory (it will say it has been removed in the top left area of screen).
*Go to the shrine, ACTIVATE a different quest besides the Umbra one, and talk to the statue. When the dog starts to talk, fast travel to the Imperial City Water District.
*Get the Umbra sword before you activate the Shrine in the first place, then talk to the Kajiit (the kitty person at the shrine standing there) before you talk to Clavicus Vile. Although anyone who is reading this has probably already started the Daedric Quest before getting the sword, so this one is useless to most people.
Sorry there were other hokey methods I tried out, but I am so tired, I forgot some of them. If NONE of these methods work for you just search around on the internet, using key words Clavicus Vile, Umbra, and game freeze.
Well folks, hope everyone finds what best works for their system with this glitch. I have heard of LOT'S of people getting around this glitch. Don't sit around and wait for a patch. You just have to see which way works best for you! Good Luck