How do I share my wireless connection via a wired cable?

This is surprisingly easy at least with Ubuntu 12.04. I have not personally been able to test this on 12.10 but It has been confirmed that this works there too. I use the Gnome Classic desktop but similar steps should work with any desktop.

I'm connecting my Windows 7 machine to the internet via the Wifi on my Ubuntu 12.04 machine.

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