phpmyadmin is nice, but not good for backup-restore functions. It F's up at regular intervals.
Instead, issue the mysqldump command on your server (through telnet for example) to get a properly formatted dump.
Then use that output as input for your new database.
If you do a mysql-connect to the 'original' database from the 'new' server, you can do this:
mysqldump database -h <database_host>
-u <username> -p --add-drop-table | mysql <database>
this will connect to the remote server (source) and do a dump of database 'database' then it will pipe all output to your local mysql server and execute it on database 'database'. Et presto, an instant copy of your database.