I would check the Sygate Forum, but I think you would disable DHCP on the router, and put it in router instead of Gateway mode.
Sygate would handle DHCP and NAT, the Linksys IP (make sure it is outside the DHCP scope of the Sygate at a fixed IP) would be the IP given for outside DNS services.
But I would check as I have never set it up this way.
I see no reason it would not work.