Shannon Hicks, who is a wealth of Flex Knowledge, gave me the solution for this problem. Apparently, an earlier version of the Flash Debug player incorrectly assumed debugging mode.
The solution is to use an updated Flash Debug player. You can find a suitable version inside the installation for Flex Builder 3. Mine was located here:
C:\Program Files\Adobe\Flex Builder 3\Player\win
I am using Flex Builder 3 Beta 2 and it works. Later beta versions of Flex Builder include the correct version of Flash Player as well.
Thanks Shannon for helping me with this. It was DRIVING ME NUTS!