Your current database should have some sort of "export" function. Use it to export the tables that make up the database in SQL format (what it does is create SQL instructions to create the tables and then instructions to input the data).
Then go to cPanel -> PHPMyAdmin, go to your new database and you should see an "Import" link. Click on that and use the page that appears to import the SQL dump from the old database.
Please note: If you export the database and include instructions to create the database (instead of just the tables), you will get an error. Make sure you only export the table structure/data.