There is "no suggestion that Mark Harper knowingly employed an illegal immigrant", Downing Street have said.
The Immigration Minister has announced his resigned over the issue.
Number 10 said David Cameron had "accepted his resignation with regret".
More top news
"Important clarity" or the grounds for the "theft of democracy"? The PM's declared aims for negotiations have received a mixed response.
A divided theme across the UK today
400 people will run for mental health charity Heads Together, which was set up by William, Harry and Kate.