How can I make that a site automagically show a nice "Currently Offline" page when the server is down (I mean, the full server is down and the request can't reach IIS)
Changing the DNS manually is not an option.
Edit: I'm looking to some kind of DNS trick to redirect to other server in case the main server is down. I can make permanent changes to the DNS, but not manually as the server goes down.
I have used the uptime services at DNSMadeEasy to great success. In effect, they set the DNS TTL to a very low number (5 minutes). They take care of pinging your server.
In the event of outage, DNS queries get directed to the secondary IP. An excellent option for a "warm spare" in small shops with limited DNS requirements. I've used them for 3 years with not a single minute of downtime.
EDIT:
This allows for geographically redundant failover, which the NLB solution proposed does not address. If the network connection is down, both servers in a standard NLB configuration will be unreachable.