It not yet known what caused the death of the baby rhino, who was born in January and named Fara.
She became ill on Tuesday morning and although keepers and vets stayed by her side, Fara died in the afternoon.
"It is too early for us to speculate to the cause of Fara's death but everything possible was done to help her," said Dr Nick Davies, assistant curator of mammals.
"It is a truly sad time for us all."
Officials have decided to close the rhino house today to give her mother "some quiet time".
Fara arrived on 31 January to mum Kitani, and her birth was caught on film.
The female calf landed safely in a "textbook" birth, said the zoo's curator of mammals Tim Rowlands at the time.
It was the first time Fara's father 15-year-old Sammy had sired a calf.
Eastern black rhino are a critically endangered species and the zoo says there are just 650 left in the wild.
"Fara was the fourth calf born here in the last three years and to date we have had a very successful breeding programme so Fara's death will be felt deeply by all those who cared for her every day," said Dr Davies.
The zoo says its breeding programme is using "ground-breaking science" to monitor the hormone levels of female rhinos, ensuring they are introduced to potential partners at the best time.
It also provides support to other organisations around the world who are involved in protecting the survival of the species.
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A black rhino calf has died at Chester Zoo, keepers have announced.