Using a cut length of 3/8 polished rod ($4 from Tractor Supply) as an insert, the bottom wedges nicely into the bottom taper of the antenna body (the base still attached to the car), but the top is loose.
A few turns of 1/4 inch conducting copper tape (amazon.com) snugs up the fit. Shown below is the copper tape before being pushed up into the antenna body.
The upper part of the antenna is now easily removable if need be by pulling the rod from the base, and the repair is virtually invisible. Reception is as it was before.