Here's what I'd do, then:
1) Turn off DHCP server in your Linksys. You don't need it, & with only two computers you might as well just hardcode them anyway.
2) Set your Linksys's LAN address to 192.168.1.1, subnet mask of 255.255.255.0.
3) Set computer #1 to an IP address of 192.168.1.2, subnet mask of 255.255.255.0. Also, go to the default gateway tab & configure a default gateway of 192.168.1.1. Go to the DNS tab, & hardcode your ISPs DNS servers.
4) Set computer #1 to an IP address of 192.168.1.3, subnet mask of 255.255.255.0. Also, go to the default gateway tab & configure a default gateway of 192.168.1.1. Go to the DNS tab, & hardcode your ISPs DNS servers.
5) Reboot both machines & call it good.
Viper GTS