Picture of Replacing a broken auto antenna
My Saturn Vue was in a tangle with a mailbox. In the process the antenna was snapped off. The thoughtful people at Saturn supplied the antenna mount with a shear bolt but forgot to make both ends removable. The replacement antenna base costs around $50. If you have all of the tools, this repair costs less than $2.00

I will show you how to replace the antenna without replacing the base.

UPDATE: Be sure to remove the antenna before going through the car wash.
I took the car to Saturn to get a free oil change. They wash every car after service. When the car came out it had no antenna. They replaced it with the correct factory antenna and base.
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Step 1: Prepare the base

Picture of Prepare the base
Prepare the antenna base by first drilling a hole. Use the appropriate bit for the size tap and bolt you are going to use. Drill slowly, wear safety glasses. Metal shavings in the eye are not at all fun.

Step 2: Prepare the bolt

Picture of Prepare the bolt
Next put two (2) nuts on a long bolt and thread the bolt into the hole you just prepared.
Tighten the first nut hand-tight down against the antenna base.
Then tighten the second nut against the first. Turn them tightly against each other to lock them in place.

Remove the bolt from the base and put two more nuts onto the bolt. repeat the tightening process. Be sure to tighten them against each other, but do not tighten them more than hand tight against the first set of nuts.

Once the second set are very secure, loosen and then move the first set out of the way.

Measure the depth of the bolt socket (after removing the remains of the shear bolt) with a piece of wire or another bolt. Mark the first bolt adding the depth to the length of the socket you already marked with the two nuts. And cut.