Hi Serge,
depends on your mailclient. In case its outlook, go to tools, options, mailsetup, mailretrieval ( or kind a like ) and tell your outlook to download both mailbody and attachments completely ( you will have to do so for every folder ).
Than create a new mailaccount pointing to the new server and simply copy the stuff over.
Alternatively you can use your local store ( pst file ).
Hope that helps, cheers,
Ralph