Update: the problem is still here. I'm running Activity Monitor and it is showing Mail consuming 90 to 115 percent of the machine (love dual core accounting) for extended periods – say 10 plus minutes. There is lots of free memory (over 1 GB).
When Mail is "frozen" quitting and restarting helps. The IMAP server is responsive if I check e-mail from my iPhone, but in Apple Mail the Connection Doctor window is very slow to respond (over a minute for the green dots to show up).