I’ve seen this before if you have multiple IP addresses on the router. You have to be careful to make sure that any private IP addresses are being NAT’d so in case your router is trying to use one of its private address as its source IP, it has a way of getting out to the internet.
I have done this. All websites are replying to ping. For example: ping nu.nl, or ping tut.by etc… works. Ping comes back. And when I press STOP, I see the real IP address of the host.
The router is functioning now only as an access point, and has a PPTP connection to my home (so I can control it). So it has a dynamic IP on the PPTP link.
But the routing table shows 0.0.0.0/24 gateway 192.168.5.1 (local ADSL router).