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Police describe the case as "heartbreaking"
The boy - called Charlie by nurses - was left by his mother who police suspect gave a false name.
The woman was last seen walking towards the car park at Newham General Hospital in east London on Wednesday.
Charlie's mother, believed to be from Birmingham and aged 25, went into labour outside a shop in East Ham on Wednesday evening and a passer-by called an ambulance.
She was taken to hospital and gave birth just minutes after arriving.
Just before midnight "Miss Sctarra" went outside the hospital to have a cigarette and then disappeared.
She is described as eastern European, possibly Romanian, and slim with blue or green eyes, shoulder-length blonde hair and is about 5ft 4in tall.
She was wearing a grey bomber jacket with a pink zip and a multi-striped hooded jumper when she went missing.
She spoke broken English with an eastern European accent.
Detective Sergeant Gerard Healy from the Newham Missing Persons Unit said: "This is a heartbreaking case and we hope that Charlie's mother comes forward to be reunited with her baby boy.
"Whilst Charlie is receiving the best possible care, we are concerned that his mother may require medical attention and support.
"We urge anyone who has seen her or been in contact with her to contact police."
Anyone with information can call police on 020 7275 5728.