What you'll need to do is remove the glass by heating carefully with a heat gun or hair dryer. The glass is glued in, so if you apply enough heat and work slowly and carefully, you can sometimes remove the glass without breaking it. Others have also said to soak the mirror in paint thinner to dissolve the glue. If you do break the glass, you can get a replacement made from an auto glass shop, of course you will lose the date code of the original glass, but it's not the end of the world unless you're out to win the Mopar Nationals or something.
Once the glass is out you will see two swedges or rivets holding the mirror plate onto the ball stud that allows the mirror to pivot. These are what have worn out causing the mirror to loosen up. Drill out these old rivets and use sheet metal screws in their place. You can now adjust the tension of the pivot by tightening or loosening the screws until the tension is where it needs to be. Reinstall the glass using the appropriate adhesive and you're good to go.