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1. Upload new .asp content (still using hard coded links)
2. Leave old .html content on server until next update, then redirect to new content (for other SE's)
What pitfalls should I look out for?
Depending upon what your old and new URLs are, it may even allow you to simply setup a redirect from the old to the new. This won't transfer your PR to the new URL (Google won't see them as the same link), but it will at least keep your old URLs in the index, and keep any backlinks you may have from breaking.
I've heard before that pages could be ranked higher the longer they've been in the index. I personally don't believe this but is it true?