At first, I figured I would probably just have to go find another $20 Playstation on eBay or Craigslist. However, that turned out to be unnecessary when I found this third-party replacement online for right about $5 including shipping. This Intec G7220 power cord is a polarized, 6 foot long 110 volt cable that is UL listed. Mine came shipped coiled up and twist-tied together, and placed in a manila envelope. It appears to be about the same thickness and similar quality to the original Sony Playstation 2 cord. I plugged it in, fired up the Playstation and it's been working as expected.
Note that when I was looking for a cord online, many retailers had this cord listed as being compatible with the PS one and the PS2 Slimline console. This is completely incorrect, and if those retailers had even glanced at the back of a slim console for half a second they would realize they have entirely different connectors.
This cord only works with the original Playstation and Playstation 2, not the slim versions.
The original models of the two systems have the power transformers built into the system, and take this same standard cord. Note that this cord can also be used in a Sega Dreamcast as well as many household stereo systems. It's a pretty universal design where the end that plugs into the console looks like a small square next to a small circle. Slim models of both systems have the power brick transformers in the cord (similar to a laptop computer), and the end that plugs into the console is a single small round plug that looks similar to a headphone jack.