The drama follows a slew of dramatic interest in the heist, after two films and a radio play about how the thieves broke into the vault.
Larry Lamb and Phil Daniels starred in a big screen version released in April, while Michael Caine and Ray Winstone will star in another upcoming film.
A cartoon also appeared on the BBC News website last year.
Six men were jailed for up to seven years for stealing items from the vault in central London in 2015, with an estimated two thirds of the valuables still unrecovered.
The robbery took place in London's diamond district, carried out by a gang of career criminals who drilled three 25cm holes through a concrete wall to climb into the vault.
Spall, who won best actor at the 2014 Cannes Film Festival for Mr Turner, will star alongside Kenneth Cranham in four 60-minute episodes.
Jeff Pope, ITV Studios' head of factual drama, said: "This is one of the most high-profile crimes of the last decade and we wanted to understand what had happened - and why it had happened.
"The research threw up some fascinating detail and blew away many of the misconceptions about this story. It was not about a bunch of 'loveable old blokes', many box holders lost everything in the raid and we will reflect this.
"But the planning was clever and the characters involved unique."
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Timothy Spall is to star in ITV's dramatisation of the Â£25m Hatton Garden safety deposit box jewellery raid.