Graphics: Icon Editor Needed for Transparent Background of Shortcut & Favicon

I can make a desktop shortcut icon or browser favicon.ico from a .jpg, but I can't seem to get the background pixels to be transparent.  How do I make the Desktop Icon Transparent?

Can someone recommend the "Best" FREE icon editor?  What's your favorite?  

I use Photoshop and some of the pixels have no color, just transparent background.  But I can't save to .ico from PS.  I save to .jpg for example.  I've tried Icon Constructor, but can't seem to make those non-color pixels transparent.
How, for example, do I use Icon Constructor to make the non-color pixels transparent?  


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      Here are some I tried
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IconConstructor
     This one looks like the best, but costs $25 - $99 after 15-day trial
     http://www.canadiancontent.net/tech/download/Icon_Constructor_-_advanced_icon_creator.html
     http://www.iconconstructor.com/default.htm
     -  I still don't see how to make the pixels transparent

Snap Files Icon Sushi    ( Free )
    http://www.snapfiles.com/opinions/icon_sushi/icon_sushi.html

PixelToolbox    ( Free )
     This one's OK, you can see every pixel
     http://www.axiomx.com/PixelToolbox/

TransparentGUI    ( Free )
     http://www.geocities.com/mangokun/transparentgui/transparentgui.htm

AW Icons Editor Pro  ( $39 )
     45-day Trial
     http://www.download.com/AWicons-Editor-Pro/3640-2195_4-10255646.html?pn=2&sb=0&v=0

List of Icon Editors for Sale
     http://www.iconcool.com/download.htm

BrotherSoft.com
     Assorted Programs of all types
     http://www.brothersoft.com/downloads/icon-editor.html

Not Good:
Active Icon Studio 5
     Don't bother with this one
     http://www.brothersoft.com/desktop_utilities/icon_tools/active_iconstudio_20932.html