During a lecture with Jennifer Egan, writer and Pulitzer-Prize winner, she was telling stories about the Brooklyn Navy Yard and one of those was that at one point during the second world war, 2 ships were brought in which were both damaged beyond repair. However because each ship still had one half which was sort of usable, it was decided to cut both of them in half then weld them together again. An amazing story, and probably correct, but I couldn't find any reference for it and keep on wondering how this was done practically as it seems quite challenging. (I do not have a link to back this story up, but I heard it just last night so it's still fresh in memory, and Egan is more than notable enough for me.)
From TWO INTO ONE Shipbuilding and Shipping Record 28 June 1945:
The U.S. destroyer escort Menges is back in service again. But only two thirds of her is the original ship; the other third was U.S.S. Holder another destroyer escort. Both ships were badly damaged in the Mediterranean, the Menges by two torpedoes, which killed 30 men and destroyed a large part of her stern. They were both docked at Brooklyn Navy Yard, where the Holder's stern was removed and grafted on to the Menges.
and photos of the repair process here: http://www.navsource.org/archives/06/320.htm
Another incident, which while not during WWII involved more famous and more substantial ships, was that the bow of the unfinished Iowa-class battleship USS Kentucky was grafted onto the USS Wisconsin after Wisconsin collided with a destroyer.