Never use a bad power supply. Here is a list of good ones.
http://www.tomshardware.com/forum/id-2547993/psu-tier-list.html
Points in favor of your Power supply:
1. ASUS may have shipped it as the power supply in one of their desktop pcs.
Points against your power supply:
1. Very old design with high 3.3V and 5V output and only 216W of 12V. 12V is used by Video and CPU today. 216w of 12V is much less that the label 350W output. (output figures from the sticker on the PSU as seen in the photo at Amazon.)
2. Price on EBAY is $20-$25. Not very many good $20 PSUs out there.
Net, I'd drop $40-80 on a good PSU.
Now for your actual question:
Definitely use the adapter if the power supply is any good. For example, an...