being that it is that old, it most likely uses the "large waist balance staff", the Waltham part number for the large waist is 1364, and the small waist is 1365, if you look at Cas-kers, I believe they have images with their staffs, so you can compare it to the one your trying to replace.
Here is a link for Cas-kers, do a search for the waltham part number, I got a return on the 1365 staff, check the image to see if it matches your staff, if not you will need to do some searching for the 1364 large waist staff, check Daves Watches, fleabay or Otto Frie might have it also (I list Otto Frie last as they are pricey)
http://www.jewelerssupplies.com/shop-bi ... e71cd4468d