Thanks so much for posting that link Steve. I hadn't thought of that great ally of the vintage car owner: commonality of OEM suppliers. It's saved me more than once with my French cars.
Sure enough, a search of the web revealed the early Kienzle mechanism clock, called Type 1 in the piece you posted, is found on Mercedes, Porsche and BMW vehicles up to somewhere in the early 70's when the quartz oscillator clocks, labeled "Quarz-Zeit" on the face, appeared. My '66 1800 has a Kienzle drive and the '73 coupe and '76 Si both have a quartz mechanism.
It's very simple to disassemble the clock to get to the point where you have to remove the hands in order to remove the face, which is held by two small screws. In order to remove them without the risk of bending you need to apply equal lifting force from both sides. Best tool to use is a pair of spoons. Small, espresso spoons would have worked better but this is a German car so strudel spoons were called into action. This technique works equally well to remove VDO speedo, tach and other gauge needles, which are much tougher to remove without damage.
A very tiny screw holds the time setting knob:
...and upon removal of the face, this is the view:
Now, turning to the back, there are 4 screws on the corners holding the back plate to the clock (two removed screws visible in this photo) as well as the ground connection to the back plate, emerging from the circuit board that lurks beneath:
This is where solder wick comes in real handy:
I used a 25 W soldering iron, 30 or more would have been much quicker. After wicking away, or otherwise removing as much solder as you can, the three remaining screws can be removed...
...but you might find out that the minute amount of solder left in place will still be holding the back down. So just apply the soldering iron at the same time you are pulling the back away and it will pop off. This pulling-while-heating might have worked from the start without using any solder wick but you will see later where you might really need the solder wick.
It's off!
Small phenolic disk surrounds the ground post laying on top of a foam separator. Remove both and you have a clear view of the back of the circuit board.
The circuit board contains two electrolytic capacitors that are very likely the problem if the clock isn't running at all. They are rated at 16V, which leaves little safety margin. Radio Shack quality caps are at least 35V nowadays. Both caps in my clock were 47 uF (microfarads) and I happened to have two in my junk drawer (clean living!:grin

but you can find them at Radio Shack. If you look closely you might even see some evidence of leakage from the old caps. Just desolder and replace the capacitors with new ones and you should be done. Here are my old caps:
For extra credit, I wanted to remove the circuit board off the clock gearing and that's when I got into trouble. Besides the obvious screw on the PCB, the electric motor terminal pins are the last thing holding the board to the mechanism, they are the two solder points identified here:
From below, you can see both pins and the very thin wire that emerges from the motor winding and wraps around the pins:
If you try to just apply the soldering iron to those solder joints and pull as you heat, odds are you will pull off the winding wire and snap it off the motor. Use the wick. I didn't at first and one of the wires snapped off and it was the one that is at the inner, or beginning, of the winding core. The only way of getting at it and reattaching it to the terminal pin was to crack open the motor, unwind the angel hair wire onto another temporary spool and then rewind it back onto the core. This wire is incredibly thin, 0.08 mm, to be exact.
So... If you are reading this post, and it is the year 2041 and all E3 clocks have been extinct for years, you also might want to attempt this repair...:roll: For posterity, here is the abridged photo essay:
Before you attempt to solder the wire, don't forget to sand it with 800 grit or smaller since it's enamel-coated.
Motor back in place. At the clock factory, it is riveted with very small rivets to the plastic gearing base, but something as 'rednecky' as twisted up teflon tape (strong and with a little bit of stretch) works wonderfully to secure it back in place. Gore dental floss would have been as good, or better.
My clock has been running with under a minute a week discrepancy without any adjustment, but there is trimmer cap on the circuit board, it's the component on the left with the screw on top. The new caps are seen on this photo.
And finally, it works!! Current consumption is around 8-10 milliamps, BTW.
http://www.youtube.com/watch?v=h6VOsVVs2mo