Switching servers (Quick question)

Greetings. I’m switching web hosts in the near future. I’m currently running WP 1.5. If I understand correctly I should be able to do a database dump, put it on the new server, and then upload my WP files in the same directory that they were on the old server (domain will stay the same. New sever uses Apache like old one).

Will this work? Can I upload fresh files downloaded from www.remarpro.com or do I need to download all the files off my old server and throw them on the new one? (ignoring template stuff).