1: Take the image you want to place on the sphere, and turn the image into a symbol.

2: Make an open path that is a half circle.
(You can use the circle tool to make a circle, and use the rectangle tool to subtract that down the center of the circle. Then use the direct select tool to delete the diameter.)

3: Select your new path and go under effect > 3D > revolve.
(A pop up window will reveal a cube and several options. Move the window so you can see the path you created earlier. Click the preview check box to activate it and now you have what appears to be a sphere. Now on the right side of the window should be an option that says map art. Click that option to get another window. Now at the top of this window,  close to the center there should be a set of numbers that look like page viewing options. Click on the arrow pointing to the right until the view window looks like a bell curve. Now there should be a drop down menu in the upper right corner of this window that should have none selected right now. Click in there to import your symbol. At the bottom of the window should be a box that, if checked, will make the sphere invisible but will leave your symbol visible. Now you will have to use this window and the original 3D window to adjust the symbol to the rotation you like.)

4: Close all windows, look at what you did and be happy.