Bernard Lemaire's puzzle: dissecting a cross to a square

In June 2002, Bernard Lemaire sent me a picture of the following cross, and said that he had found a 5-piece dissection of it to a square.

Five pieces, huh? How hard can that be? If Bernard had not told me five pieces, I would probably have settled for a dissection with more pieces. But since Bernard had said five pieces, it was a matter of honneur to find it myself! It took me about an hour. The solution is nifty and appears deceptively easy by hindsight.
See what you can do with the puzzle. If you get stuck, take a look at a hint.
Be sure to check the photo gallery to see Bernard.

Copyright 2002, Greg N. Frederickson.