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You will need a flash decompiler program, some of which can be found on google, but it may not be looked upon as 'proper'.
Just opening the file in Macromedia Flash MX for example, will not work because the keyframes, motion tweens and general animations are not broken up into editable layers that you can just remove.
If you have access to the native .fla file, you can download Flash MX trial version, open the .fla file, edit then export back into .swf for your site.
If you purchase In a Flash Pro, (which I think you are using) then you will be able to remove this limitation to your .swf output files and be able to remove their ad when you create new files. This is common practice for trial versions of certain software, of which I am sure you are aware.