Have you installed "Internet Connection Sharing" in the Host Windows ME PC? It is a bit more complicated than in XP (where you simply tick the box "Allow other network users...etc") in that you have to install the ICS protocol.
Go into "Control Panel > Add/Remove Programs > Windows Setup > Internet Tools > Internet Connection Sharing and tick the box. (That's actually in Win 98SE because I don't have ME installed but it should be very similar.) Then, follow the on-screen instructions but, as with XP, the Host PC's network adapter will be given the IP address 192.168.0.1 and the Client should be set to get its address automatically.
As usual, any firewall software needs to be configured to allow access to the IP addresses of the networked computers and, if using Zone Alarm in the "Host" PC, its Internet Zone Security level needs to be reduced from High to Medium or ICS traffic will be blocked.