While discussing the sssihl migration, where they have an edu GApps account and hence can use GApps IMAP migration tool, S mentioned that some other institutions like MedTrust had similar requirements. But they're using the Standard (Free) Edition.
For just a single account to another account, the migration would be possible, though slow, by just adding both accounts as IMAP accounts on a desktop client like thunderbird, making sure that emails are retrieved completely and not only headers, and dragging and dropping from one account to the other, as given in this discussion.
In the link above, the popular answer is the way it's being done with the IMAP migration tool on the Edu version. But the lifehacker link just below that gives this other method.