A stupid 301 redirect question

[Armadillo]

I have one web site with two domain names pointing to it.
One name is new, never been touched by a bot.
The old name (that I plan to phase out) has a PR1 on Google.

I have a 301 redirect on the old server while the dns changes propagate. The old server will be taken down after dns is updated.

Stupid question-
When the old site is gone, should I put a 301 on the new site as well?
This would "redirect" visitors using the old domain to the new domain, which are already at the same place!
Only the name is different.
This seems like a waste of server resources, but I want to keep Google happy.

[g1smd]

Yes, add the redirect. Showing the same content at multiple URLs is a Duplicate Content problem.

You'll need to redirect anything that is not exactly www.example.com to www.example.com for any URL with wrong domain name, non-www on any domain, and any request (even www.example.com) that includes a port number or trailing period on the hostname value.