I'm not familiar with this as I've never had to do it, but I'm pretty certain it is possible to do this in phpMyAdmin, here is a tutorial I've found, haven't looked through it, but I guess it should give you all the relevant information...
Easiest way would be to download the files through an ftp client such as FileZilla and download the databases (if any), then re-upload to your new address, if I have understood your question right.