i take it Internet Connection Sharing is turned on, on the WinME box if you have other machines sharing the net connection.?
Drop to a prompt and type 'neat' this will open up the graphical network config utility. Click on your ethernet card connection and click 'edit'.
Change your gateway to the IP of the WinME machine.
This works for me (I am using a Win2K) box though. It should be the same.