The appointment, which follows open competition, will take place immediately.
Mr Bull, who succeeds Ian Stewart, has been the acting Director in BFI since May this year.
Bull worked at the National Audit Office from 1979-1996, and has been a consultant to the Swedish International Development Authority. He was educated at the University College of Wales, Aberystwyth.
The BFI is an independent unit within the Department of Social Security set up to inspect social security benefits administration and counter-fraud activity within DSS agencies and local authorities, and to promote good practice.
Cotton seedling freezes to death as Chang'e-4 shuts down for the Moon's 14-day lunar night
Fortnite easily out-earns PUBG, Assassin's Creed Odyssey and Red Dead Redemption 2 in 2018
Meteor showers as a service will be visible for about 100 kilometres in all directions
Saturn's rings only formed in the past 100 million years, suggests analysis of Cassini space probe data
New findings contradict conventional belief that Saturn's rings were formed along with the planet about 4.5 billion years ago