Thanks for posting the information. With reference to what you said I downloaded Allan McComb's ICSConfig and used it to configure as follows:
On Host Computer with Win Me:
I disabled the DHCP but DNS is still on. Changed IP address of machine for LAN Nic to 192.168.1.10. I am able to connect to Internet without any problem. I am able to ping LAN ip addresses as well as external IP addresses. External NIC card still has the same configuration as required by the cable modem company with a static IP address.
On Client Computers with Win Me and Win 98:
I changed it get automatic IP addressing from the addressing from the router. I used Winicfg to check and found they get the IP addresses generally 192.168.0.3 and 192.168.0.4. I am able to ping all the IP addresses in the LAN but I still am unable to connect to the internet. I am unable to ping outside also.
I am still missing something, let me know if you could suggest me anything else I should do.