Standing more than 132m (433 ft) high, the building south of John Lewis will be home to around 450 students.
It will be taller than Swansea's Meridian Tower, which is the current highest at 107m (351ft).
Council planners approved the move on Wednesday and the building is due to be completed in summer 2018.
It will have a shop or cafe at ground level, and a "sky lounge" available for hire.
A 42-storey student accommodation will be Wales' tallest building after Cardiff council planners gave it the go-ahead.