<div id='header-wrapper'>Change the no in showaddelement='no' to yes, this enables you to add gadgets on top of your header in the Page Elements customization in the future.
<b:section class='header' id='header' maxwidgets='1' showaddelement='no'>
<b:widget id='Header1' locked='true' title='Azure Haze (Header)' type='Header'/>
</b:section>
</div>
*optional*Change the number in maxwidgets='1' from 1 to any number you like if you want to setup more gadgets in the header part.
Remove the code coloured red, which is the current header you're using. Save template (you might be asked to confirm your Header1 deletion, just confirm) and head on to Page Elements.
Now you should be able to see the "Add a Gadget" box on the header part, click on it and add HTML/Javascript (if you don't see it, add the gadget from your sidebar, then shift it to the header).
Type in the <embed> code which looks like this (replace http://yourhost.com/yourflash.swf with the address of your .swf file):
<embed src="http://yourhost.com/yourflash.swf" loop="false" height="240" width="1300"></embed>
