There are two problems with your mesh that I can see.
While in Edit Mode, you can select a face along the rim and move it to see that there is another layer in the same spot.
To clean up the overlapping faces you can use the Remove Doubles tool since all the vertices are in exactly the same places. It's available in the Toolshelf and can also be accessed by pressing w to access the Specials menu.
Now you may see an big mess because some of the faces are inside out and I have backface culling disabled so I can spot errors like this.
Ctrl+n will make the normals consistent.
After recalculating the normals, the inside face of the glass is backwards. By hovering over one of it's vertices and pressing l you can select the entire piece at once as shown below. Next you can press Ctrl+f to access the Faces menu and choose Flip Normals. The previous image shows where the tool is located in the Toolshelf.
Now for the second problem that you mesh has. For the Glass Shader to function correctly, the mesh should be completely enclosed. As it is now, there is no rim and this should be filled in. You can do this easily by holding Alt and clicking the Edge. Next press and hold Alt + Shift then Click the second Edge to add to the selection. Now you can press Space and type in find the Bridge Edge Loops tool.
The black inky blobs you see are shadows and I have no idea how to clean that up without disabling shadows for the entire scene. If you figure that out, feel free to share your results. :)
I also added an Environment texture and a couple extra lights since Glass tends to look better when there are surroundings to reflect and refract. Here are the render settings I used except that I increased the Samples to 1024.
Also, the ground plane is actually two ground planes that are overlapping.
The next image shows one of the two ground planes moved down to show it's there. It's something that you should always watch out for because it will cause rendering artifacts if geometry overlaps like this. With Object Mode overlaps this is easy to fix, just delete it.