I'm sure there are easy ways to do so in FCE but as I'm not near FCP right now I'll tell you an even easier way with Quicktime Pro. Your version of QT will be QT Pro if you have Logic Studio or FC Studio. (You can also use Compressor to convert multiple files. It's included with FCS and Logic Studio).
Open the video file in Quicktime.
Choose File > Export...
Set the Export to Quicktime mov
Very importantly you should look at the size, frame rate, all settings of the movie.
There is a similar way to do this in FCE by working with each clip, but I'm not near Final Cut so can't direct you accurately.
Btw, have you checked out the Final Cut 6 101 tutorial?
It's very similar and relevant IMHO for FCE 4 users:
This may help you.