if you have have a shell (or telnet/ssh access to a machine) you can:
dump the DB:
mysqldump --add-drop-table -h hostname -u username -p dbname > dump.sql
(it will ask for the db password)
once you have the file dump.sql you can make it eat to the new db in this way
mysql -h hostname -u username -p dbname < dump.sql
(it will ask for the db password)
of course a db should be prepared in the new mysql installation (but the name havent to be identical)
its a good idea (for migration issue) having same users/dbname on both mysql, and if you can give the new one the cname (or A record) of the old one... so you dont have to rewrite/correct your applications and scripts... just wait for the dns to refresh!
if you work on local machine yo ucan omit -h hostname
if you connect remotely be sure the user you are using can connect remotely to mysql
hope it helps