It's better to use static ip addresses in this situation. You don't say what os you are using on the 1st box. Basically you need to place the second box on a different subnet to the 1st. In this case you have the primary addresses on 192.168.0.X so you could use 192.168.1.X for the second range. You would need to tell the router to forward requests for port 80 ( conventional www access) to the ip address of the Internet server. The next step is to get forwarding going from the first box to the second. i.e. you need to tell the 1st box to forward requests on port 80 to the second box. You would have to work this out for yourself based on the os.The next thing is gateway addresses. The gateway for the server is the ip address of the ethernet card the server is attached to on the 1st box and the 1st box uses the router address as the gateway.
Pay particular attention to security of the server with regard to a fire wall and ssh access. If you don't need access the server from outside your own lan only have port 80 open to reduce the risk of compromise. There are lots of bad people out there that are going to be probing the server for holes from the minute you put it on line. Make sure you read the security recommendations in the security section of LQ. You may think that your server does not contain anything important and you don't need anything fancy but it could easily be hijacked and used to attack other servers out there so please be responsible in your approach.
I also have some basic info on adsl based servers with links to other info on my server