If the old one is still up and running, start ADE on it, and hit ctrl-shift-D. (cmd-shift-D if it was a Mac).
Otherwise, you are stuck until you run out of authorizations, at which point you can contact Abobe to have them all reset.
Thank you for your quick reply. I already gave the computer away - therefore I think I have to wait until the authorizations run out.
If Adobe resets all authorizations what happens to the files I saved in ADE? Are they deleted then?
No the files stay there. You reauthorize the device with the authorization it had before, and everything works as before on the devices you reauthorize.
You just now have more authorizations available on that ID for new devices.