Export as an H.264 file with AAC Audio, as quicktime conversion. Then change the file extension to .flv. This works flawlessly! If you will be doing this alot I would recommend the el Gato TurboHD, its a USB drive you plug into your computer and contains a program that pumps out H.264 files very fast! I use it all the time, all of my editing online and put them into flash this way.
Will that keep the audio, I was able to successfully change the mov to a flv but when i did that i lost the audio, i will try exactly what you said and see if that works, if it does i will be soo happy, been trying to figure this out since yesterday.
You could export directly to FLV with Perian (http://perian.org/#download). It will integrate FLV and many other omitted formats to your Quicktime. In FCP, under Export using Quicktime Conversion, you'll have the choice to export to FLV.