One easy way around this is to draw a "matte" outside the bounds of your artboard, covering up the "offscreen" artwork. If the browser forces the SWF area to be larger than you've specified, the user will just see this matte.
In your case, since the site has a plain black background, you could just draw a frame of four big black rectangles. You can group the rectangles and then hide them so it's easy to continue editing the offscreen items -- just unhide the group before you publish a new copy of the project to the web.
A second option: if you're going to bring your project into Flash Builder, you can add a snippet of code that will make the application rescale to fill the larger zoomed size, rather than simply "uncropping" areas outside the stage. Just add this code to the root <s:Application> tag in your Main.mxml:
preinitialize = "systemManager.stage.scaleMode = StageScaleMode.EXACT_FIT"
Hope that helps!
Peter, I tried the second option you proposed and it enlarged just the top right hand portion, about an 1/8th of the whole screen.
It scaled to the window every which way, but only that portion.
Could you post a screenshot of what you're seeing?
Also, did you make any other changes to the code after you brough the project into Builder? Or did you make any changes to the HTML "wrapper" file that displays the SWF?