As I understand it, there end up 4 different kinds of cubes in the bag.
8 corners (3 black), 12 edges (2 black), 6 faces (1 black), and one center.
Each must go to its assigned spot type, odds are 1!6!8!12!/27!.
Now on to orientations of each cube. Each cube has 24 potential orientations (if we assign a specific direction for front, top, etc). (Pick a front, then pick one of the 4 adjacent to be top)
Corners will be correct in 3 orientations for a 1-in-8 chance.
Edges will be correct for 2 orientations (1-in-12).
Faces only have to have the proper face pointed outward (1-in-6).
The Center cube isn't visible, rendering its orientation meaningless.
Combining all this, the odds become slightly less than one in 5.465 trillion trillion trillion. (5 followed by 36 digits)
EDIT: Actually calculating the chance, and not just the denominator, I get 1.8298 x 10^-37. The answer above is just a different way of stating the same value.