I have just got a new ATI Radeon X1650 Pro 512MB

The power supply should be efficient enough. You have all the drivers downloaded, so I would suggest upgrading your memory. If you only have 1 gig of ram, half of it is going straight to video card.
I also suggest downloading "glary utilities" program which is free. It will boost and clean registry, remove trojans, spyware, duplicate files, adware, manage start up programs, and so much more.

Also when running games, it is a good idea to shut off any other running programs. Every one that will allow you to.
Just not enough memory in the system.
I would recommend 2.5 or more.

I doubt it's your power supply, unless you have a very low quality one. It could be that your supply is defective, however.

You should check to make sure that your games are DX9 compatible, because if you're running in DX10 mode, that may cause incompatibility problems. After all, you're running some relatively old equipment. I don't think AGP is even supported by new boards.

I would recommend downloading a registry cleaner called CCleaner, and just keep running the registry clean function until no more errors come up. If you upgraded your operating system, it may not have installed (or uninstalled) your NET frameworks properly, and so it could be referencing different versions of the same software simultaneously, leading to a crash. A registry cleaner will clear up a lot of the bugs associated with uninstalls.

If that fails, go into your BIOS on computer startup, and poke around. It's possible that your computer doesn't have some features enabled, such as an external graphics card.

If you're overclocking the card, you should stop. It may cause it to become unstable.

If none of these work, I would recommend mailing the card back in to whomever you purchased it from. Your card should have a manufacturers warranty of at least a year. If you have been overclocking, your warranty is null and void on the party anyway, but you can still give it a shot.

System could be overheating make sure nothing is stopping the fans from turning ,also try with the case open.