This French favorite has been gaining notice here thanks to the charming 2001 French film Amelie; it entered the American popularity list in 2002 and has been rising ever since. Also spelled Amalie, the pronunciation is like Emily but with a short 'a' sound at the beginning.
There's definitely a lot of confusion over the roots of, and differences among, all these related names, but Emily, Emma, and Amelia all derive from different roots and have different meanings, and Amelie is a descendant of the Amelia branch.
Amelie is currently enjoying great popularity in the U.K. (It's Number 51 in England), Number 94 in Ireland and Number 13 in Germany.
Probably inspired by the name of French King Louis Philippe's consort Marie-Amelie, Balzac used the name for two of his prominent characters.