This can be done by more than one way. First way, you can try to dump-to-sql the whole DB, then create a new DB and import the sql-dump in the new DB. For this, you may use mysqlhotcopy or mysqldump MySQL utilities.
Second way, you can also use the RENAME TABLE command by creating the new database then use RENAME dbname.table TO newdbname.table. You can list multiple tables like below:
RENAME TABLE current_db.table1 TO other_db.table1,
current_db.table2 TO other_db.table2,
current_db.table3 TO other_db.table3;