You don't need a black sphere around your scene, the world settings do the exact same thing.
Then it is not the color on the material of the object, but the environment that you need to change.
By default blender will have the environment set to a gray tone, preventing dark shadows (think of it like trying to get dark shadows on an overcast day, the light of the environment will make the shadows less dense).
Making the Environment completely black will mean that only the lights in the scene will illuminate the objects, there will be no ambient light.
But why are you not seeing the black sphere but the gray world as background? is really the question.
When the camera is looking at areas that have no objects you will see the gray environment as background for the scene.
Most likely the sphere is at a distance that is farther than the clip distance of the camera, so it is being ignored, and what you see instead is the color of the background set in the world settings.
To understand clip distance read: How do I increase the render distance?