Chucks are actually pretty cheap. Harbor Freight has many 3 and 4 jaw chucks.
One of my more recent purchases there, is a small 3 jaw self-centering chuck mounted on a MT2, which they sell for their mini lathes. I have it in my manual woodworking lathe right now -- I enjoy making small projects.
Anyway, I believe it can be separated from the mount, and the price is definitely right...I got it on sale for $9.99. Runout is in the neighborhood of .003, on my example. WAY closer than is necessary for most woodworking.
Let me see if I can find a URL for it.
Their other offerings are equally interesting, especially the bare 4 jaw chucks for the woodworking lathes. This one mounts on a 3/4" threaded shaft -- talk about EASY!
Or, if you want to make one yourself, you might try just duplicating the second design...it's not difficult. I think I've seen someone's site, where they did just that.
Since you said it'd just be 1" round stock, you might try a collet. It'd be a 1" diameter hole bored into something, some radial slits to allow it to compress, and a nut around it to compress it onto the workpiece... It'd work just like the collets on your router. Should be simple to make...cheap too.
-- Chuck Knight