Trevor Noah was hardly a household name in the United States when he was chosen to replace Jon Stewart on Comedy Central’s “The Daily Show.” Audiences have been slow to embrace the South African-born comedian in the role, and ratings for the show are down. After Noah’s debut in September 2015, Newsday’s Verne Gay summed up the reluctance many viewers felt: “We subliminally expect to see Stewart sitting at that nice new desk. . . . Yet there sat someone we hardly knew.”

Americans will know Trevor Noah much better after reading his terrific new memoir, “Born a Crime: Stories from a South African Childhood.” Not that the book is in any way a promotional tool for the television show. Nor is it the conventionally thin gruel that constitutes a celebrity memoir these days. Noah has a real tale to tell, and he tells it well — the tale of a boyhood in South Africa during and after apartheid.

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Noah, born in Johannesburg in 1984, is the son of a black woman and a white man — she a devout Christian from the Xhosa tribe; he a Swiss chef and restaurateur. The Immorality Act of 1927, reproduced in the book’s opening pages, prohibited “illicit carnal intercourse between Europeans and natives”; hence the book’s title. His parents’ relationship was unorthodox through and through; Patricia approached Robert, a neighbor and friend, about having a child with her, but she didn’t want to get married and she didn’t require him to play a role in the child’s life. (Robert did end up playing a minimal one.)

“If you ask my mother whether she ever considered the ramifications of having a mixed child under apartheid, she will say no,” Noah writes. “She wanted to do something, figured out a way to do it, and then she did it.” Patricia Nombuyiselo Noah emerges as the most vivid character in “Born a Crime”: an iron-willed, independent-minded woman determined to raise her son for great things — and not in the least reluctant to discipline him when necessary, as it often was.

Among the many virtues of “Born a Crime” is a frank and telling portrait of life in South Africa during the 1980s and ’90s. Noah has written the book for an audience he assumes knows little about his country, and he lays out clearly the situation of blacks — corralled into segregated townships, limited to employment as maids or factory workers (women) and miners or farmhands (men), subject to curfews and other restrictions of a police state.

“Born a Crime” is not a chronological narrative but a collection of autobiographical essays that feel like developments out of Noah’s comedy routines. A chapter on extreme churchgoing — Noah’s mother dragged him to services four nights a week, plus three on Sunday — features an episode of hitchhiking and ends with Noah and his Mom jumping from a moving car. (A little scary, but trust me — it’s funny.) Another chapter, about life at his grandmother’s house in Soweto turns on the lack of indoor plumbing (a universal fact in Soweto then) and sees 5-year-old Trevor quietly going potty on a newspaper on the kitchen floor under the unseeing eyes of his blind great-grandmother — who nevertheless retains her sense of smell. (Again — trust me.)

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Occasionally, the book’s loose structure works to its detriment, and some chapters feel more developed than others. Short explanatory passages between chapters feel unnecessary, since so much background about South African society is woven seamlessly throughout. But these are quibbles. “Born a Crime” offers Americans a second introduction to Trevor Noah, and he makes a real impression.