1999 Mercedes Benz ML430 clicking sound

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This is a common problem on C classes but can happen to other models as well. What you are experiencing is a broken stepper motor armature. This armature transfers movement from this small motor to the blend doors of your ac system. You will notice this mostly at start up as the vehicle does a self test at this time. This is a big repair and often times requires full removal of the entire dash. Typically 4-6 hours of labor but the part is usually under 20 bucks.

what should I expect to pay for the repair? Is this something I can do myself? I am mechanically inclined.

Depending on the shop you should expect to pay anywhere from $400- $600 for the repair. The part to replace is cheap (below 20 bucks) but this is a very labor intensive job. If you are familiar with MB dash disassembly and complete removal then you could try doing this on your own, however from my experience because of the amount of work involved and the complexity of reassembly I would highly recommend bringing this to a reliable shop.