How to fade a video clip in Adobe Flash CS4?

I have a .mov video file I've encoded into a .flv format and imported into the Flash library.  I've placed an instance on the stage.  How can I make the video, while it is playing, fade out?  I've played around with all the tweening capabilities and it seems I can only get it to fade out if I convert it to a "movie" symbol and create a tween, changing the alpha from 100 to 0 at the end of the tween.  That works fine, but the video does not play during the fade.  The video does not actually play during the fade out, just the first frame is displayed throughout the fade.  In actuality a new instance of the video movie symbol is added to the stage, rather than implementing the fade at some point in the middle of the original video on the stage.  Any help is MOST welcome.