Setup your new server with Exchange as part of the current exchange environment. All you have to do from here is create a new mailbox store with matching policies. Then migrate mailboxes from your old server to the new server. For public folders, set replication between the two servers.
Once all mailboxes are migrated and all public folders have been replicated and the mailboxes work, change the MX record to your new server. If you have another Domain controller, you can power down your old server and make sure everything is working. Then you can uninstall or rebuild your old exchange server.