The couple have reportedly named their newborn daughter Kaidence Donda West, according to US website MediaTakeOut.com.
Her first name keeps the Kardashian family tradition of beginning with a K, while her middle name is said to be a tribute to Yeezy's late mother, Dr Donda West.
They said of the labour: "This was not planned. This came on very quickly. The whole family was dispersed all over the place!"
The source added: "[the baby] is healthy and Kim is doing well...There's a great sense of relief. Everyone is very happy for her and Kim is relieved [because] this was a very stressful time."
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