Flight Record is a paid iOS App (€2,39) that allows you to record video from the AR.Drone without the need to Jailbreak your phone. It appeared in the App Store yesterday. I have not tested it yet, but it appears that the On Screen Controls remain invisible in the recordings judging from the screen shots in the App Store. The app records video in the H264 QuickTime format at 15 frames per second. Your iOS device needs to be running iOS 4.1 in order to run the app.
Be warned: since the app is also used to control the Ar.Drone and it is not an official Parrot application some caution should be taken when using it.
Below is a demo video that was recorded with the App.