Basically Mike Koval's answer will do the job, that is setting wmode="transparent"
. All other HTML/CSS/JS things will remain the same from your own link.
Full embedding code HTML version (by swfObject's generator) will be:
It will be always better to use swfObject in JS, something like:
var flashvars = {};
var params = {};
params.wmode = "transparent";
var attributes = {};
swfobject.embedSWF("path/to/flash.swf", "id-of-the-div-the-flash-will-go", "800", "600", "9.0.0", false, flashvars, params, attributes);