If your driving license has been issued in a non-EU country and you plan on living in Germany longer than a year, you will need a German driver's license (Führerschein). Whether you can simply exchange yours or take a part or all of the driving exam, only depends on where your driving license was acquired.
If you have a US driving license you might want to check How to Germany and the AmCham websites for details, as it depends on the state in which your license has been issued and the actual German regulation.
In Leipzig the motor vehicle registrar is the Ordnungsamt Kfz-Zulassungs-, Führerschein-, Melde- und Passbehörde (Abteilung). A motor vehicle, or car/auto is das Kraftfahrzeug, Kfz in short. It is now located on Prager Str as mentioned on the above website. On that internet page you can find numerous documents necessary for you car registration to download.
And the last link of that page leads you to the online reservation system of the motor vehicle registrar.... very practical to avoid very long queuing time!
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