Old site --> phpMyAdmin --> go to the database you want, angus_foo
Click on Export (top of screen)
Use the default settings [ OR if your DB is large, use "Custom" and compress it, leaving all the other settings the same ]
Click GO, it downloads to your computer
New Site --> Create the empty database using cPanel -> mySQLDatabases , say it is named schmoo_bar
Now go to phpMyAdmin , select schmoo_bar, then Import (top of screen).
Select file you downloaded above, click "Go" and wait while it uploads and fills your new DB.
You have to create a new DB user in mySQLdatabases and add him to schmoo_bar with ALL PRIVILEGES. Write down the password.