Sounds like the wiper motor park circuit switch. Is that the original motor? Also, we need video of you trying to make the wipers stop.

Try electrical contact cleaner on switch (NAPA has spray can) . Also It will clean and remove moisture on motors connectors. Also soak down the motor it wont hurt the windings or stator surfaces.  
 

Last edited by Don (4/30/2014 11:27 PM)

     Every time a contact opens up an electrical arch starts to develop across the opening. This happens until the contact opens far enough and the arch goes away. Over time that arch starts to damage the contact. Probably what happens is that arch partially welded your contact shut. After some time the spring tension in the contact broke that weld and the contact opened up. This is when your wipers shut off. Let's hope it has nothing to do with water or you have bigger problems.
     I agree with the guys that it is probably in park switch but this could also happen in the on/off switch as well. This will probably rear its ugly head again. If it does (this check is for a 2 speed motor) there are 2 green wires on the wiper motor that go to the park switch. Pull of one of those wires off. If the wipers stop, the problem is in the park switch. If they continue to run it's the on/off switch. If it’s a water problem call a plumber.
 

Last edited by wsinsle (5/01/2014 5:56 AM)

Bullet Bob wrote:

Please provide more information.

Is your car a fastback or hardtop, what year, what engine/trans/rearend ratio, disc or drum brakes, standard or delux interior, interior color, body color, plain or tinted windows, does it have air conditioning, wheel/tire size, and most importantly....has ANYONE been messing with it lately?

BB

OK, Bullet... You nailed me on that one !!!!!
Dang... I ask ONE QUESTION in 15 years of having this forum and I forget the rules...

I have a two-speed wiper setup I have been hoarding for about 20 years now, so I guess it is finally time to upgrade my single speed setup to dual speed.  I also still have one of the intermittent wiper kits that I used to sell.  It will park the wipers for you even if you kill the ignition with the wipers in operation.