I am looking for a load balancing solution that is extremely reliable. People talk a lot about hardware failures, but I've also encountered situations where routing to entire data centers has been negatively affected.

Is there a load balancing or cloud solution that uses a single IP that can be automatically rerouted to different geographically-based instance according to network availability?

Thanks

I think with single IPs that will be hard and leaves you still open to routing issues.
During the .com bubble times people would use "traffic directors" for that (iPivot, F5 3DNS).
Technically you would have delegations in DNS to 2-3 traffic directors. Those would continually check availability of loadbalancers in different datacenters, and the RTT + load of the servers in the datacenter, and then create a dns reply to the client based on the fastest available site.

I just don't know what the current state of art for that would be :(

I came across this myself a while back. I haven't tried it, but it seems you basically need to get your own provider independent IP address (or beg your provider to borrow their IP), an Autonomous System Number, and permission to use the upstream router, so you can update it via BGP.

http://content.ruud.org/bgphints/articles/anycast.html

This one details someones attempt at actually trying it with EC2 ELBs.

http://ajohnstone.com/achives/high-availability-across-multiple-data-centers-multihoming-and-ec2/

Global Directors from F5 could be used, but too expensive. If you have a budget, you can put bunch of them into each data center and load balance on the data centers level.