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Meghan has previously said the book, written with rhyming text, began as a Father's Day poem she wrote for Harry shortly after Archie was born in 2019.
In a handwritten-like note inside the book, Meghan wrote: "For the man and the boy who make my heart go pump-pump".
Meghan also narrates an audio version of the book, which features watercolour illustrations by artist Christian Robinson.
Harry and Meghan, a former actress from the US, quit royal duties last year, leaving Britain for California, where they now live.