[EDIT: This post was typed with the idea you'd build it yourself]
It's possible, but cutting it close.
The problem with your old computer:
1) Your mobo and CPU are too old/slow
2) mobos that old have older RAM. You cannot keep this, and must by new stuff.
3) The Case for your old mobo might not be good for a new PC (old PCs needed little cooling, if anything. New ones need side vents over the CPU, and at least 1-2 other fans).
4) video card.
5) HD will be IDE, and slow. Not to mention has little storage capacity. Might look into upgrading to a SATA drive. Much faster read/write and mroe storage.
Going for the Intel chipset, only because I know these parts exist:
ASRock 775Dual-VSTA (cost $50 on NewEgg). Takes older 533MHz FSB chips and still allows upgrade to 1033MHz FSB chips later (including Conroe).
CPU: Here's the thing. You can get an OLD chip (cheap) and run it until you can afford a better one. Conroe prices will be dropping like rocks in a few months. I know for a fact that you can get a Celeron 2.56GHz 533MHz FSB chip for $50 on NewEgg. I just bought my sister one a couple of months ago. It is called "Intel Celeron D 331 Prescott 2.66GHz LGA775 64-bit processor w/ Execute Disable Bit Model _______" on the invoice slip. Price: $58.
This motherboard allows either DDR1 or DDR2, AND either AGP/PCIe. I've got this one now.
You need at LEAST 512MB ram, bare minimum these days. 1 GB better but costs twice as much. Can get them for about $50/512 or $100/1024.
POWER SUPPLY (caps for effect). Your old PSU can't support newer chips and vid cards. Especially considering the low amp requirements on older PCs. You need at least a 400W. I went with a nice, low-cost, but high-quality $45-or-so FSP Group PSU.
Video: You can keep your old card and upgrade later if you like. This mobo listed has both AGP and PCIe slots. I went with the best PCIe budget card there is. The Ge7600GT is one of the best cards out there, price-for-performance, and I got an eVGA "7600GTS" -- a GS card clocked at GT speeds (has a GS bios). Basically a GT but only cost $140. PCIe of course
Hard drive: Sata 1.5 120GB for $70. Can it get any better?
CASE: You've got a PSU, don't rely on one included with the case. Make sure it's full ATX, mid tower recommended, and has a side vent (or two). I got mine for $30 (no lie), but it's no longer being sold. It's black with a fire engine red cherry front. Now they only have plain black/beige of the same type (shame, really). The thing with cases is to get a CHEAP case. They range from $30 to $200. Go for the $30 side.
So if you recycle monitor, keyboard, floppy, CD-ROMs and burners, and all that stuff, you come down to:
Mobo: $50
CPU: (cheap/decent) $60/$200
RAM (512/1024): $50/$100
PSU: $40
Case: $30-40
Hard Drive: $70
Video Card: $140 (down to $100 for a low-end card)
So you come to
$120 for Case/Mobo/PSU alone
$230 for a Celeron CPU and only 512MB RAM, reusing 3D and HD
$370 for a decent CPU (Pent 4 3.0GHZ area) 512MB reusing 3D/HD
$630 for a decent CPU, 1GB DDR1/2, new HD and new 3D card.
Not counting price fluctuations and the like, that leaves some leeway for other things. Also allows some leeway if you want a better case, a better PSU, vid card, etc etc.