hmmm theres a possible number of causes for your issue...
We'll start with the simple stuff....
1) have you double checked that your system is indeed using your Radeon card and not a possible onboard graphics chip that might be hiding there? You'd be amazed how many times people have fallen for it and left the onboard enabled which isnt sufficent for the latest games.
2) have you tried the latest drivers for your radeon card (click here)? Or did the problem only start occuring when you installed a version different to the ones you previous had?
And now some things to consider:
3) When you say the pc reboots are you sure? Or are you getting "VPU Recovery" error messages?
I only ask this because I've seen Radeon cards give black screens during a game and then a "VPU Recovery" error message appears saying that the GRAPHICS CARD (not the pc) has been restarted and that the graphics card is now running in software accelerated mode......thus giving the illusion to the untrained eye that the pc has rebooted. This may explain why you get further problems after the first occurance of the day.
4) Have you been attempting to overclock your card?
If you can get your hands on one....you might wanna try using a different Radeon card (preferably same model, and use the same drivers) in your machine, this will soon tell you if the card itself is knackered.
If your system is only 6 months old, and your graphics card was purchased from new, then you should still have it under warranty. speak to your supplier.