Ms Begum, a teenager who fled the UK to join Islamic State in Syria, gave birth to the boy in a refugee camp in the middle of February, having already lost two children.
The Syrian Democratic Forces, a US backed, anti Islamic State rebel group which controls the camp where Ms Begum is living, confirmed the baby’s death.
A paramedic had earlier told the BBC that the infant, who was about two weeks old, had been suffering from a lung infection and died on Thursday.
Ms Begum was 15 when she and two other schoolgirls went to join the terror group in February 2015.
She resurfaced heavily-pregnant in a refugee camp in northern Syria last month and spoke of her desire to return to the UK, as the self-styled caliphate collapsed.
On February 17, her family announced the boy’s birth and said they believed he was “in good health”.
Home Secretary Sajid Javid stripped the British citizenship of Ms Begum - who is from Bethnal Green in east London - amid a fierce national debate over whether she should be allowed to return.
Her family, who pledged to appeal against the decision, also wrote to Mr Javid pleading with him to allow a safe passage for the boy to come to the UK.
Last month, Mr Javid confirmed the boy was a British citizen and said he had considered the child’s interest when deciding to revoke Ms Begum’s citizenship.
Asked whether there was any plan for Ms Begum’s son, Mr Javid told the Commons Home Affairs Committee it would be “incredibly difficult” for the Government to facilitate the return of a child from Syria.
“If it is possible somehow for a British child to be brought to a place where there is a British consular presence, the closest place - it might be Turkey for example - in those circumstances I guess potentially it is possible to arrange for some sort of help with the consent of the parent,” he added.