For InnoDB, the following seems to work: create the new empty database, then rename each table in turn into the new database:
RENAME TABLE old_db.table TO new_db.table;
You will need to adjust the permissions after that.
For scripting in a shell, you can do this:
mysql -u username -ppassword old_db -sNe 'show tables' | while read table; \
do mysql -u username -ppassword -sNe "rename table old_db.$table to new_db.$table"; done
Notes: there is no space between the option
-p and the password. If your database has no password, remove the
-u username -ppassword part.
Also, if you have stored procedures, you can copy them afterwards:
mysqldump -R old_db | mysql new_db