As the manual says in Copying Databases you can pipe the dump directly into the mysql client:
mysqldump db_name | mysql new_db_name
If you're using MyISAM you could copy the files, but I wouldn't recommend it. It's a bit dodgy.
Integrated from various good other answers
Both mysqldump and mysql commands accept options for setting connection details (and much more), like:
mysqldump -u <user name> --password=<pwd> <original db> | mysql -u <user name> -p <new db> Example : mysqldump -u root -pmypassword old_database | mysql -u root -pmypassowrd new_database