jmraspa,

The old Schwinns used a headset size that was eventually adopted by the original BMX crowd. They are 1" headsets but have different dimensions. If you can find what is commonly refered to as "OLD SCHOOL BMX" sized headset, it will work. But they can be expensive.

I used this Chris King on a Super Sport. It was overkill at $100+ but the bike had a new paint job and lots of nice pieces on it, so was worth the investment. It would be extreem on a Varsity.

Try typing "Schwinn Headset" in the search engine at ebay. You'll find plenty. The ones for the Krates and Stingrays will work.