It's cheap & easy to do on an old Schwinn. Save the original headcups ONLY, or if thy're thrashed, get yourself one of these headsets:
http://porkchopbmx.com/vp-h755-bmx-bicycle-headset-1-threaded-w-32-5mm-cups-chrome/
ALL you're going to use from the 1" threaded headset if the cups. Everything else, save for parts or whatever.
Next, you're going to want to get a cheap bazz-turd loose ball 1.125" threadless set. Best are the loose-ball NECO-branded headsets; it will also work with the seemingly identical ones sold under the "Sunlite" label. Here's one I have used in the past:
http://www.ebay.com/itm/Bicycle-Threadless-Headset-1-1-8-x30mmx27mm-Chrome-BMX-MTB-Road-Cruiser-Bike/262575445427?_trksid=p2047675.c100009.m1982&_trkparms=aid=888007&algo=DISC.MBE&ao=1&asc=38636&meid=87ab4d4c7efd4cff9eb94482c275609f&pid=100009&rk=1&rkt=2&sd=291735463472
(Yes, it matters which looseball 1.125" threadless headset you use; some have oversize lower-races, or sealed bearings or other stuff that may make them incompatible with the cups.)
OK. Now, you're going to need to use EVERY part of the 1.125" threadless headset, except for (obviously) the cups, and the caged bearings. The reason for this is, while both headsets use 5/32" bearings, they use a different number of balls in the cage... if you try to run the caged bearings, it will bind and you will mangle the retainers. What you'll want to do is run loose, no cage, straight-up commando-style loose ballz in there. I forget what the count is, but basically, you want to use some very sticky number2 grease, and pack the race with enough balls to fill it all, minus one ball. So, leave one ball's worth of "gap." Easiest way to do it without dropping balls is, flip the frame over and pack the lower race first; drop the fork in and hold it tight enough to the headtube that no balls spill out. Flip the bike rightside up, and then pack the top race the same way.... lots of tacky grease, and extra room for exactly one ball. Finish installing the top part of the headset and stem, and test everything out.... I don't know if you already mess with threadless headsts, but there are lots of good tutorials online (like, on the parktools site) that will tell you how to set the preload on threadless 'sets...
One final note: for best results, you'll remove any crown race that might be installed on the fork, and swap it out for the crown race that came with the 1.125" threadless headset you used for the rest of the conversion.
I hope this helps; i've done this a few times, and it's worked awesome for me. Cageless loose balls are a PITA for maintenance, though. Another option would be to hit up RRB member
@chattymatty and ask him to turn down a sealed Tange LAV-82 for a Chicago Schwinn. He can polish the headset at the same time for ya; I went that route for one of my Worksman, and the sealed bearings are entirely worth the small amount of added expense...
Regards,
Rob