To get a severely disabled person’s ID card, you need to apply to your regional social service authority (Versorgungsamt). You’ll find the address if you search online for “Versorgungsamt”.
Alternatively, you can ask at the citizens’ information office (Bürgeramt) of your local authority or consult the local authority’s website. There is also an online list here giving the addresses of the social service authorities all over Germany.
The ID cards are usually issued for a limited period of time. In most cases they are valid for between two and five years.
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