If you are looking for something to game on with that budget, and are opposed to building, I suggest spending ~$320 on a pre-built HP or similar (sales are your best friend. Read all the ads in the Sunday papers and shop around until you find something good. Don't rush it. If you can hold out till Black Friday, you can get an incredible deal. Also, open box, scratch-and dent, and clearance could get you a deal). Spend the remaining $80 or so on a decent graphics card. Something along the lines of a 4670 should be sufficient for light to medium gaming, though something with DX11 support would be a plus in 5 years (assuming you keep it for as long as your last one). Again, look for sales on a video card, though I suggest buying this online. Sites like newegg and tigerdirect can save you a ton of money. Open box deals are great as well if you are pressed for cash.
If you have any questions, or would like recommendations, please feel free to ask.
BTW, avoid budget laptops for gaming. All but the lightest games will be unplayable.
EDIT: If you go the route of upgrading the video card, make sure you get a full-width tower. That means this
, not this
. If you go with a slimline/mini, it will become incredibly difficult or impossible to find a decent graphics card that fits. Also, make sure it has a PCI express x16 slot. Any full-sized desktop should, but after my last Dell, nothing would surprise me.