You found the eBooks under My Digital Edition under the My Documents folder. Move this folder on the new Device on the same Place.
Thanks for the answer Nanaky. That didn't work. It's possible that it didn't work because one of my computers is a Mac, the other is a PC laptop. The Mac has the bookshelf definitions in it, but it is also an older version of Digital Editions.
In this case try only move the ePub files of the folder. This files can you import new in Digital Edition.