Hello. I am seeing a very similar problem. I'm running an Arduino Duemilanove that is talking to a "Processing" app on my Windows 7 machine. I have run the combined application for DAYS without any comm problems. However, if I STOP the Processing app, the Arduino resets itself (over and over). If I restart the Processing app, all is well again. This happens regardless of where the power comes from.
With the app running normally, if I simply disconnect the USB cable, the Arduino is happy and does not scram. This tells me the problem is coming from the PC, but I don't know how to prevent the resets on the Arduino side.
I see this as a big problem in my particular application. I hope a solution comes soon.