Mr Williamson turned down the Prime Minister's offer to resign before he was sacked (Picture: PA).
In her letter to Mr Williamson, Prime Minister Theresa May said: "Thank you for your time this evening. We discussed the investigation into the unauthorised disclosure of information from the National Security Council meeting on 23 April.
"This is an extremely serious matter, and a deeply disappointing one.
"It is vital for the operation of good government and for the UK's national interest in some of the most sensitive and important areas that the members of the NSC - from our Armed Forces, our Security and Intelligence Agencies, and the most senior level of Government - are able to have frank and detailed discussions in full confidence that the advice and analysis provided is not discussed or divulged beyond that trusted environment."
The letter continues: "That is why I commissioned the Cabinet Secretary to establish an investigation into the unprecedented leak from the NSC meeting last week, and why I expected everyone connected to it - Ministers and officials alike - to comply with it fully. You undertook to do so.
"I am therefore concerned by the manner in which you have engaged with this investigation. It has been conducted fairly, with the full co-operation of other NSC attendees.
"They have all answered questions, engaged properly, provided as much information as possible to assist with the investigation, and encouraged their staff to do the same. Your conduct has not been of the same standard as others.
"In our meeting this evening, I put to you the latest information from the investigation, which provides compelling evidence suggesting your responsibility for the unauthorised disclosure.
"No other, credible version of events to explain this leak has been identified."