Police have said that the 52-year-old driver has been arrested on suspicion of causing death by dangerous driving.
An elderly woman who was a pedestrian out in the town centre has been confirmed dead.
Chief Inspector Graham Milne, has said: "Obviously this is a tragic incident and I would like to extend my condolences to the deceased's family.
“I would also ask anyone who was in the area at the time and witnessed the collision to contact us on 101."
It is believed that the bus pulled away from a stop several yards away before it careered across the road and crashed into the bank window.
Witnesses have said that three people became trapped under the Arriva bus and several pedestrians were knocked down.
One told the Northern Echo: “It was absolutely shocking, I’m still shaking now.
“That person was just out doing a bit of shopping and now they’re dead”
Air ambulance, fire fighters, police and paramedics were called to the scene around 11am this morning.
The roads around the scene have been closed and the Great North Air Ambulance said there was one patient with serious injuries, but not thought to be life threatening.
One took to Twitter to praise the emergency services for their efforts.
We will bring you more updates on this sad news as it happens.