To confirm: the lights stay bright when you attempt to start it? If they do, the battery's probably OK.

If I recall correctly, I had a similar problem that I corrected by replacing the electrical portion of the key switch. Volvo was quite clever with this car: you can replace the electrical portion of the switch without taking apart the key part, and you thus don't need to drill out bolts, have special anti-theft-bolt-removing tools, etc. I believe there's a discussion on Bay 13 or here - can't recall - on how to do this. It's an afternoon's work or less, half of which will be looking for where the last fastener is hiding on a component.

Another possible problem is something in the starter. Can you hear the solenoid click when you turn the key, or is all silence? If you can hear the click, but nothing else happens, it's probably starter replacement time. If no click, you likely have a problem that power's not getting to the solenoid, which takes me back to the previous paragraph.