Well, now that I’ve taken a look at the Inspiron 530s, I can see its price point is pretty darned low for the basic model. That’s usually how it is with the big box retailers: their cheap models are dirt cheap, but their high-end machines cost $800 to $1000 more than it would to simply buy the components yourself. They really overcharge you on the upgrades.
However, one problem you would encounter with the Inspiron: it comes in a very slim case, which is possible because they all use onboard graphics chipsets, meaning no graphics expansion cards. Chances are, you would not be able to fit a standard video card into that case. There are low-profile PCI-e cards, but that would limit your field of options quite a bit.
Anyway, I’m going to give them a run for their money. Here’s my first go, trying to match their specs as closely as possible:
CPU: Intel Celeron E1400 2.0GHz Dual-core processor
Motherboard: ECS LGA 775 Intel 945GC Micro ATX motherboard
Memory: 2 * 512MB Kingston DDR2-667 Dual Channel SDRAM
Hard drive: 250GB Seagate Barracuda 7200RPM
Optical drive: LITE-ON DVD burner
Case/PSU combo: Linkworld MicroATX mini-tower case with 430W PSU
Total price on Newegg: $230.76
This package is actually somewhat better than the baseline Dell Inspiron 530s, since it has a dual-core Celeron instead of a single-core. Of course, the rest of the parts are the same sort of cheap, lowest-bidder crap you will get with a Dell. I would strongly recommend against buying such a system, and instead pushing up the budget just a bit into the $400-ish range. You’ll be able to get much nicer components overall, and you’ll be saving even more money over a comparable Dell machine.
Of course, the one thing I can’t beat them on is the bundled operating system, but that’s really a given in the low end machines, since Microsoft has such a vested interest in ensuring that the bargain basement PCs from the major retailers all ship with a copy of Windows. If you happen to have a license for your old machine and you won’t be using it anymore, that would probably work fine. Or you could pony up $90 for an OEM copy of Vista or XP Home Edition.
Personally, I would recommend starting with the parts I suggested in my post above. You could actually scale down to a cheaper CPU if you want, like the Celeron I listed above; it probably won’t make a huge difference for your usage scenario, and it’s $60 cheaper. You could probably take off another $30 or so by dropping to a cheaper graphics card, but it’s worth pointing out that the 8600GT at $85 (or this one, at $70) is going to be leagues beyond the cheap Radeon that Dell charges you $60 for.
Then toss in a hard drive, some RAM (scratch the DDR2-800; DDR2-667 will be good enough, and cheaper) a good cheap PSU, and a roomy mid tower case. Since silence is a priority for you, a slightly larger case would probably be helpful: the roominess and improved ventilation will help keep your devices at comfortable temperatures even with passive coolers. I would still recommend a pair of quiet 120mm fans, though. (Or heck, maybe a Celeron will be fine in stagnant air with a good enough cooler. I just know I wouldn’t feel entirely comfortable doing that myself. I’ve never built a machine that didn’t require serious cooling just to stay stable. I’m running a Core 2 Duo E8400 overclocked to 4.05GHz right now. 
I suppose you could try it yourself and monitor the core temperatures, and add a fan or two if it gets too hot.)
