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I think Amd uses less power its according to the processor being used.
A miniITX build, as ilikecake was referring to, would meet your latter two requirements. (Depending on your budget, you may or may not hit the "cheap" criteria.)
A sample build would be...
$70 - Intel BOXD945GCLF/Intel Atom processor 230 Combo (ilikecake's recommendation)
$37 - A-DATA 2GB DDR2 667 single DIMM (free shipping)
$39 - Western Digital 80GB SATA HDD
(optical drive optional)
$56 - APEX MI-100 Mini-ITX tower with 250W PSU (free shipping)
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$202 - Subtotal (not including shipping, taxes, or rebates)
... though you may also want to check out Logic Supply and Mini-Box.com for more miniITX parts.