For large operations like this it is better to go to command line. phpMyAdmin gets tricky when lots of data is involved because there are all sorts of timeouts in PHP that can trip it up.
If you can SSH into both servers, then you can do a sequence like the following:
Log in to server1 (your current server) and dump the table to a file using "mysqldump" --- mysqldump --add-drop-table -uSQLUSER -pPASSWORD -h
SQLSERVERDOMAIN DBNAME TABLENAME > BACKUPFILE
Do a secure copy of that file from server1 to server2 using "scp" ---
scp BACKUPFILE USER@SERVER2DOMAIN:FOLDERNAME
Log out of server 1
Log into server 2 (your new server) and import that file into the new DB using "mysql" --- mysql -uSQLUSER -pPASSWORD DBNAME < BACKUPFILE
You will need to replace the UPPERCASE text with your own info. Just ask in the comments if you don't know where to find any of these.
It is worthwhile getting to know some of these command line tricks if you will be doing this sort of admin from time to time.