Among the questions from pupils at Oasis Academy Brightstowe was one from 15-year-old Seema who wants to be the first female Afghan astronaut.
Major Peake told her it was a "wonderful idea and ambition" and that she should "follow her dreams"
The link up started at 14:20 GMT and lasted 10 minutes with the school using the special radio call sign GB1OAB.
A computer programme tracks the ISS as it travels across the sky and this controls where the aerials are pointing and what frequency the VHF/UHF radio is on
As the students ask their questions the operator switches the transmitter on. At the end of the question he or she says 'over' and the transmitter is switched off
The reply is received from space and played over loudspeakers in the hall
There was a loud cheer from pupils, staff and parents as his voice crackled through the loudspeakers in the school hall for the first time and the live picture appeared on TV screens.
Sixteen-year-old Ashleigh wanted to know how many days' supplies the space station has on board and what would happen if a restocking mission could not take place?
Major Peake told him a few missions to restock had had problems but the space station had six-month's worth of reserves.
And nine-year-old Jacob asked why he had become an astronaut..
He replied he had been a pilot and it was the "pinnacle of his career" to "explore new frontiers".
The link-up was part of the Amateur Radio on the International Space Station project which aims to inspire students,  to pursue interests and careers in science, technology, engineering and maths through an interest in amateur radio.
Students from a Bristol school have spoken to astronaut Tim Peake aboard the International Space Station.