One of the force’s four companies, where extremism is said to be the most rife, will be dissolved and not replaced, the minister said.
“Anyone who turns out to be a right-wing extremist has no place in the Bundeswehr and must leave it,” she told German radio.
KSK operations will be moved to other units as far as possible, and it will not take part in international exercises and missions until further notice.
Ms Kramp-Karrenbauer said the latest findings – including the disappearance of 48,000 rounds of ammunition and 62kg (137lb) of explosives – as “disturbing” and “alarming”.
An internal investigation is due to determine whether those items were stolen or are missing due to sloppy bookkeeping.
The unit was founded in 1996, and has some 1,000 soldiers trained for crisis situations such as freeing hostages abroad, which had not been possible until then without assistance from other countries’ forces.
The military’s problem with far-right supporters emerged in 2017.
Inspections were ordered on all military barracks when Nazi-era memorabilia was found at two of them. Many of those suspected of far-right links are thought to be sympathetic to Germany’s main opposition AfD party.