Import e-mail via IMAP into GMail?

It's not supported by Gmail natively. However, since Gmail provides IMAP service itself, you can set up a desktop mail client (e.g. Thunderbird or Outlook) for both old and new accounts, then simply drag and drop all messages to Gmail's inbox.

There are also tools like imapsync which can copy everything automatically. (Although I am unsure about how they'd handle Gmail's unusual IMAP structure, so I'd probably stick to manual copying.)

Gmail does not support importing mail via IMAP. However if you are a Google Apps user, then that Gmail does support it.

When using Google Apps Gmail you don't get Import by default. Your administrator must enable this explicitly.

https://support.google.com/a/answer/2525613?hl=en