Emergency services were called to St Vigeans Road in Arbroath just before 3pm on Saturday.
The baby girl was taken to Ninewells Hospital before being transferred to the Royal Hospital for Sick Children in Edinburgh.
It is understood she is in a critical condition.
Her injuries are being treated as unexplained with officers currently at the scene while investigations are being carried out.
A Police Scotland spokesman said: “A six-month-old female child is in a critical condition in the Royal Hospital for Sick Children in Edinburgh.
“The serious injuries she has sustained are as yet unexplained.
“Enquiries are at an early stage.
“The investigation is being carried out in joint partnership with Angus Council Social Work Department and the NHS.
“As part of this investigation, there will be a police presence in St Vigeans Road, Arbroath.”
A Scottish Ambulance Service spokesman added: “We dispatched one ambulance to the scene.
“We transported one infant to Ninewells Hospital and the patient was then transferred to the Royal Hospital for Sick Children.”-