I've been using Final Cut Pro for years, but never been asked to capture an entire tape and have the software automatically cut it into single clips at every camera stop point.
I checked the manual and couldn't find any information on it.
I know that iMovie does it, so it follows that the much more professional package should be able to. Does anyone know if final Cut Pro will automatically cut the capture into its separate shots while capturing?

Yes it does. "DV Start/Stop Detect" under the "Mark" menu. Works with footage caught thru FireWire with FireWire deck control on. Footage caught any other way, I don't know. And it marks it, doesn't cut the footage.

I'm confused, every time i've ever captured any footage it has done exactly what you are asking. Or am I reading your post wrong?