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"In late 2014, Ms [Rachel] MacGregor - Mr Craig's former press secretary - approached me making various allegations about Mr Craig,'' he wrote.
"Some of the allegations were serious and I decided to pass these allegations on to others within the Conservative Party and ultimately to Whale Oil."
"I am now aware that a number of statements I made to others about Mr Craig were not true."
Exactly how much was paid by Williams to Craig has not been revealed in the statement.
Williams said he deeply regretted what had happened; as well as his involvement in spreading those allegations.
"I apologise to Mr Craig and his family unreservedly."
Williams said he would not be continuing his legal proceedings against Craig.
He said he had personally apologised to Craig and had agreed to make a payment to him - which Craig had accepted.
"He has accepted my apology and so has agreed to end his legal proceedings against me."
Craig said he was happy that the matter had been brought to a conclusion; saying that it now allowed both parties to move on.