The 17-year-old, who smashed at speed into the back of another car that had halted on the track, was trapped in the wreckage for more than 90 minutes before being extricated and flown to hospital.
"Due to the injuries sustained and despite every effort, Billy has sadly had both lower legs amputated," the FIA sanctioned junior series, the first step on the single-seater ladder after karting, said in a statement.
"He has been kept in an induced coma while undergoing surgery, from which he is expected to be woken in the next few days."
A crowd-funding site was set up to raise funds for the teenager's medical treatment and recovery, with Formula One drivers joining the motorsport community in expressing support on Twitter.
"Thoughts and prayers are with you and your family," said Mercedes' triple world champion Lewis Hamilton.
"Just watched the crash of @BillyMonger and can only say that life is so cruel...let's all help this young lad," commented former F1 racer Johnny Herbert, who suffered severe foot injuries in a junior series crash at Brands Hatch in 1988.
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