Network Topology

I need to setup a system that is completely isolated from the rest of the network, but it still needs to have internet access. This machine needs to have ports open to the rest of the Internet for web, email and DNS. Since it will have ports open and the rest of the computers on the network should be protected from incoming connections I believe this computer should be behind a router that forwards only the ports for HTTP (80), email (25, I believe) and DNS (51 I believe). I will have to double check the ports, but that's a bit irrelevant right now.

To keep the rest of the network isolated so that this machine (if compromised) cannot access the rest of the network I believe I would need a second router to block all incoming connections. The second router's gateway would be the first router's IP.

Here's a diagram to better explain what I tried to put into words.

My question: Is this the best way to go about keeping computer 1 completely isolated from the rest of the computers on the network, while still allowing them to all share the same connection? This is not a high-budget job, obviously, so these routers are not going to be Cisco or anything exotic.

// Edit: Forgot. I also think I could put computer 1 on its own subnet, 10.x.x.x and the rest of the network on 192.x.x.x just to help keep things as best isolated as possible. Would this matter?