A new game has come out, called "Mafia II", a sequel to "Mafia".
Both Steam and Direct2Drive advertise the original game for sale, but it's literally impossible to add this game to the cart on either of those sites. Predictably, support at Steam utterly ignores all questions. Steam is the Antichrist.
I'm looking for a legit and hassle-free way to download-purchase JUST "Mafia" not a bundle

ebay?

Blame 2K Games, not Steam. Steam doesn't decide what goes on the shelves, it just keeps the merchandise in stock.

Amazon
Edit: LOL . . . the price jumped from $20 to $50

GamersGate

The computer was built in 2001, if I recall correctly. Gamersgate does indeed have it for download, but since I'm not certain it will run on this machine, I searched for and found a demo.....took a long time to find a live link to a demo.
I realize Gog.com is struggling to survive economically, but I'd appreciate seeing someone post a comprehensive list of gaming download sites. I was unaware that Gamersgate even existed until you mentioned it. Googling for gaming download sites gets you a looooong list of garbage sites with worthless junk and lots of spyware crud.
Again, thanks to everyone who suggested Gamers Gate. Sorry not to tell everyone where I finally found the demo download.....it took hours to find and download, forgot the place I eventually discovered it.

Basic list of the most popular digital distribution sites:
GOG
Steam
Impulse
Direct2Drive
Gamersgate
Additionally you have the publisher specific ones like Blizzard, EA, etc. etc.
Personally I only use GOG, Steam, and Impulse, but your mileage may vary.

It works fine.
And Gamersgate pulled it when the preorder thing began, I believe. It might be back now though. I have a boxed copy... works great.

I would also add:
DotEmu
GetGames
GamesPlanet
And GreenManGaming

Sielle's list is a pretty good one. I'd add Telltale's site to that list if you like adventure games. But I'd imagine those too would pose a problem for your present computer.
DotEmu is a GOG-like competitor that might be worth keeping an eye on to make sure you get the best deal.
Bearcat's computer isn't the best. We've got to take that into account. As for Mafia, I believe it's back now. I don't know if it's possible to purchase at the moment though.

FYI GamersGate recently pulled their version of Mafia from sale. The reason given was that the publisher had asked them to remove it while they were preparing a new build of the game that would fix some compatibility problems with newer hardware. No details were given, but seeing that the game is now back up for sale on GamersGate, it's possible that that is a new build.
If you're still concerned that it won't work on your PC, email support at gamersgate dot com and ask them if you can get a refund in the event it doesn't work. I would not be surprised if they said yes.

Thank you----- I enjoyed the Demo, so took the plunge and purchased Mafia. I expected some trouble, and predictably the "movies" stutter a little, not a big deal. I do think it has
a bug though.
If you're considering purchase of this game, you might take these experiences of mine into account:
1. The game begins with you driving at night on very dark streets, in a car that handles like a pig in mud. It can be pretty frustrating, to say the least.
2. Evidently there is NO SAVE feature! If you mess up a scenario, you've got to play the whole thing all the way through again. That's a massive FAIL.
3.Evidently the little introduction movies before a scenario begins cannot be exited out of.
If you're trying to beat a scenario for the tenth time, get ready to sit through the little "movie" again, too. Lame.
I cannot even get through the first scenario. The map shows a road across the river. An odd white building (gas station, maybe) is on the left, and the sidewalk is on the right. According to the map, there should be a turn in the road RIGHT THERE......but there is nothing but sidewalk and the river on the right!
I paid $15 for this game. It's too early to say whether or not it will turn out to be
worth that. Right now, I'd say no. Whatever you do, play the Demo first! Bear in mind that
as of August 2010, you can get this game free when you preorder the Mafia II game,
so that might be your best option.
About Gamers Gate----- I found the whole site easy to use, really painless. Prices vary quite a bit, but there are a LOT of games on there. If you want amazing bargains, stick with Gog, but Gamers Gate is worth a look if you can't get it on Gog.
Again, thank you to everyone for the great help!