Details of cows are kept on 'cattle passports'
The agency's advice to consumers is that the risk to health from the incident is low.
The 29-month-old animal was slaughtered in an abattoir in Wales in November, put into the food chain and none of the meat is now left.
BBC consumer affairs correspondent Nicola Carslaw said the farmer had been warned not to sell the calf but illegally sold it to an abattoir.
She said there was no way of knowing whether the animal had BSE but there was no previous evidence of the infection being passed from mother to calf.
She added that the cattle passport scheme to keep track of cattle movements, run by the Department for Environment, Food and Rural Affairs (Defra), should have prevented the incident.
But Defra has told the agency that the animal's passport has not been seized.
Culling backlog
Defra said this was due to a backlog of visits to farms that had offspring of BSE cattle which had built up during the foot-and-mouth crisis.
The FSA is demanding that the backlog in the culling of offspring of BSE-infected cattle is cleared immediately.
BSE in cattle is believed to be linked to the invariably fatal variant Creutzfeldt-Jacob disease in humans.
Some scientists believe the disease can be passed to humans through infected meat.
According to the UK Creutzfeldt-Jakob Disease Surveillance Unit in Edinburgh, there have been 104 vCJD deaths since 1995 and nine people have the disease at present.