As recommended by others above, use thin (watery) superglue-- not the regular superglue. Last year I got a small bottle of Loctite 420 off the big auction site; I think it ran me a little over $15 with shipping.
I believe Woodcraft stores (or go to their website) still carry "Hot Stuff" CA (cyanoacrylate-- it's super glue) glue. It's woodworker-quality CA glue. The "Hot Stuff" comes in 2 or 3 different viscosities-- get the thinnest one. When I bought a bottle a couple of years ago at Woodcraft, I think it cost me $10-- but it was a good-sized bottle (2 or 4 oz., I forget which).
You want thin, watery CA glue like Loctite 420 or Hot Stuff, because you want it to wick all the way into the crack. Regular superglue won't penetrate deep into the crack like the thinner CA glue.
That '64 KS sounds too nice not to spend the $10-$15 bucks on the right glue. Good luck on getting her fixed up.