2009 UPDATE: Well, the video I embedded below is no longer available, and now I can't find any authorized video of the memorial service! That's a shame. But you can still read the text of Prince Harry's (heartfelt and touching) remarks about his mother.
2008 UPDATE: The BBC used to offer a video of the full 2007 memorial service for Princess Diana, but the link no longer works outside the UK.
I looked around, and the best video I can find now is the one I've embedded below. It's only 5 minutes long, but it includes Prince William's reading and Prince Harry's full speech. (2009 update: Video no longer available.)
Information on TV broadcasts of the August 31 memorial service for Diana, Princess of Wales:
You can watch the memorial service for Princess Diana at the BBC site. For some reason the direct link to the video no longer works, but if you're in the UK* go here and choose the "Diana's memory celebrated" video. After that window opens, you will see another link to a video of the full ceremony. It's about an hour long.
* 2008 Update: The video now seems to work only in the UK. And if it you're in the UK and it still won't work, my apologies. It's the only link I can find to the full service.