gmailmacosmail.app
I want to compose a message, but rather than clicking "Send", I want to schedule a future delivery date and have the mail sent then without my taking action as long as the computer is still running.
I'm currently using Apple mail with Google apps configured using IMAP syncing.
The answers here using Mail Scripts by Andreas Amann no longer works on Lion or Mountain Lion. Can something be done on 10.8 or later to automate sending Apple mail?
Have you tried selecting the folder in the folder list and then using the menu entry "Use This Mailbox For -> Sent" in the Mailbox menu?
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That should do exactly what you want.
It is possible to achieve this effect using Calendar and Automator:
Open Automator and create a Calendar Alarm.
Use the tasks “New mail message” and “Send Outgoing Messages”, customize them as required, and save (⌘S). 
An event will be created in Calendar automatically. Move the event to when you want the message to be sent. 
Best Answer
If you're using Gmail and are willing to use the (I think very awesome) web interface, check out Boomerang for Gmail. It does everything you're asking for.
EDIT:
You can also use Automator to schedule (by delaying) when your message is sent.
The Pause function is very helpful, and Apple has a whole section for Mail in the library. I would save it as an application and then run it with Spotlight or Alfred or Launchbar whenever you need it.