What projection were you using for the texture? In the Default Shader this setting is on the Images tab. For the Background sphere you'll probably need to use Spherical. That might work just by itself, however...
In the case of any spherical mapping usually the *position* of the image also needs to be set to the same as whatever you're spherically mapping to. The Background sphere is at 0,0,0 by default so it may work automagically, so to speak. If not, since the Default Shader doesn't include a Position setting, you'd need to use an Image Map shader instead. If the Default Shader has some setting you want that's not present in the Image Map shader, you could just use the new Image Map shader as an input to the Color Function of the Default Shader you already have. Just make sure to specify the position to be the same as the Background Sphere. You can open the Background Sphere settings and just use the copy button to the right of the position coordinates.
- Oshyan