Those cables I said didn't exist do exist! Cables To Go makes a 24″ single-device round ATA-133 cable (thanks to Ken Campney of Campney Business Services for tracking this down).

Unfortunately they charge $15.99 each if you order from them directly, but I was able to find two resellers, ConnectXpress ($9.95 each) and PC-Cargo ($6.00 each). I placed my order with PC-Cargo; their shipping estimate was awfully high but when I actually continued with checkout it gave me much more reasonable rates; I selected 2-day air so I'm hoping they'll arrive Monday. Then I can finally finish assembling my new server!