The damning report follows comments by the chairman of the Home Affairs Select Committee, Keith Vaz, earlier this month.
Mr Vaz said the number of unresolved cases that UKBA is dealing with - more than 300,000 - is "spiralling out of control".
He said: "There are now about the same number of cases awaiting resolution by UKBA as there are people living in Iceland."
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Parents Jay and Kateri Schwandt did not want to know the sex of their latest child before his birth.
Windrush immigrants Sonia Williams and Glenda Caesar are the first two cases to be resolved by the Home Office task force.