The most common German word for “to admit”, in the sense of “to confess”, is “zugeben”. To a lesser degree the words “gestehen”, “eingestehen” and “zugestehen” are also used. The noun for “admission” (again, only in the sense of “confession”) is “Geständnis” (n.) or “Zugeständnis” (n.) but NOT “Zugabe” (f.), which essentially means “encore”:

– “Gib es zu!” – “Admit it!”
– “Ich muss zugeben, dass ich verschlafen habe.” – “I must admit that I overslept.”
– “Sie muss ihren Irrtum zugeben.” – “She has to admit to having made an error.”
– “Der Dieb hat gestanden.” – “The thief confessed.”
– “Der Dieb hat ein Geständnis abgelegt.” – “The thief made a confession.”
– “Die Band hat eine Zugabe gespielt.” – “The band played an encore.”