okay, here we go .. the trick to making a quiet PC is having good natural airflow so that you don't need a bunch of fans blowing it around. What few fans you do have must be quiet. Also, a passively cooled graphics card is important, since graphics card fans tend to be small and therefore loud.
: Antec P180
an extensive article on the P180 at SPCR. They love it.
: Seasonic S12-380 (380W)
SPCR loves this PSU also. It's supposed to be very quiet, especially at low power-draw, which your computer will be, without a high-powered graphics card. 380 watts should be enough, since you're not gaming. If not, there are 430 or even 500W versions also.
: MSI K8N Neo4-F
: AMD Athlon 64 X2 3800+
: Corsair ValueSelect 1GB
: Asus GeForce 6200TC
This thing should be great for your purpose. It has a low profile (better for airflow) and uses the comparatively-priced Radeon X300 to wipe its bottom. Here
is a review of the one by XFX, and the Asus can only be better!
Better yet, it's passively cooled.
: Thermalright XP-90
This is a very efficient heatsink.
: Aspire air ducts
Attach one end of these to an 80mm case fan and the other end to the XP-90. This will bring cool air from outside the case right onto your CPU.
: Seagate 7200.7 200GB
I'm guessing you'll need plenty of storage space, so there you go. The best way to defeat hard drive noise is to suspend the hard drive with bungee-like cables. There are several articles on passive hard drive silencing/cooling here
Add those up and it's a bit under $900, which leaves room for tax and shipping. Hope that helps!