You can copy tables from one database to another by doing:
INSERT INTO newtable SELECT * FROM olddb.oldtable
However, I'm not aware of a way to copy the whole database at once directly in mysql. You'll need to dump it to a file like Stealth suggests and then put it back into the newdatabase.
Yes I have an existing dump and I must import it in the db. But I can't find an IMPORT command in MySQL man..... Max The one with Mac OS X Server 10.4 :)