Gotta add to this. The comment about "The wiper motor is obviously working" is dead wrong. How can that assumption be made when the fuse blows instantly?
The most likely cause, which is EXTREMELY common is a shorted park switch inside the wiper motor. I've replaced dozens and dissected a couple. There is about a 2" diameter plastic gear run by a worm gear on the end of the motor shaft. On that gear are a couple of flat brass rings. Small metal fingers with little metal, ... Uhm, round things on the end are the contacts. They look like small BBs, about the diameter of a pencil lead. One of these fingers makes contact with the metal ring anytime the wipers are not all the way down. This is the park switch. It provides power to the motor between the time the wipers are turned off and when they reach the park position. At the park position, there is a gap in the metal ring to remove power to the motor.
A second ring and contact only turns on in the park position to short out the motor. This turns the motor into a generator and puts a heavy load on it to cause it to stop instantly. Without this circuit, the motor would coast until the park switch turns on again and the wipers take another swipe. Proof of a defective shorting switch is the wipers continue to run after they are turned off at the wiper switch until you hold them down. Once stopped, they will stay stopped. They will also turn off when the ignition switch is turned off.
Getting back to the shorted wiper motor, you can remove the metal cover over the plastic gear. Some are held on with screws, most have rivets that can be drilled out. Just drill through the heads, not all the way through the housing. Remove the cover slowly because it might be stuck. The metal fingers are attached to the cover along with the wiring connector. You will see that one of the contacts is bent up and touching the underside of the metal cover. I don't know when this occurs, but either that contact got hot during operation of the motor, or the motor didn't coast quite far enough to completely break the circuit, and just enough current snuck through to heat up the contact. Regardless when it got hot, the spring-loaded contact melted into the plastic gear. The next time the wipers were turned on, the gear rotated and tugged on that contact bending it, causing it to hit the cover and short out.
As proof it's the motor and not the switch, just unplug the motor. The new fuse will not blow even though the switch is still in the circuit. If in the unlikely event the fuse still blows, I would suspect a wire rubbed through and grounding out before I'd condemn a switch. In fact, I don't remember ever replacing a defective wiper switch on a LeBaron.
The repair involves replacing the wiper motor. I've never had success trying to repair the bent contact. I tried sanding off the high spots of the melted plastic gear, and filling the hole with Mopar RTV sealer, but I could never get the contact to work right. Even if I did get one to work, I would never trust it to put it back on a customer's car.
These wiper motors were used in a number of other models including the minivans, Dynastys, and Intrepids.
Saturday, November 28th, 2009 AT 3:50 PM