...
Finally, from the top menu go to Layer>Matting>Remove black matte. This will remove the thin black line that is left over because of the black background. You now have a separated object on a new layer which you can paste over a new background.
Photoshop action
Click here to download a Photoshop action that does the same thing as the steps described above. To use it, load it first in Photoshop from the Actions palette. Then open your render and alpha image. Click on the alpha image window so it's the selected document, and run the action (default hotkey assigned to it is F12).
A note on black backgrounds when rendering with Alpha channel
...
Because the second sphere is against a perfectly black background, there is a dark border left after cutting with alpha, but it can be perfectly removed using "remove black matte".