Hi,
Try deleting preferences and see if that cures the export problem.
To delete preferences open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new empty library. Reopen your old library to get back to your projects. (To reopen your old library close iMovie and go to your Movie folder or wherever you have your old iMovie library stored. Double click on the library’s icon. iMovie will open in that library.) Possibly you may need to reboot to set the changes. Now try to export to your desktop. Deleting preferences is a safe procedure that will not cause data loss or disruption to your project.
The file that disappeared from your desktop likely was a temporary file that iMovie crearted to assist in the export function. In the normal case the temporary file disappears once the export is complete, and the exported Mp4 video file will take its place. If you have a crowded desktop possibly the Mp4 file is on there but you overlooked it. If that could be your situation, check the desktop again to see if you can find the file.
To cover the bases, upon export check to confirm that you have put "Desktop" in the destination box instead of Documents or some other location. If that's the case, you can run a Spotlight search to see if you can find it.
-- Rich
