Mr Gibson, named in a highly critical district audit report on the running of the council, has been away from his desk because of illness since June.
He did not return after the screening of a television interview that month in which he was accused of withholding sensitive information from members. He was visibly shaken on the programme.
The staff sub-committee has also agreed to sack housing director John Arthur Jones, who has been charged with a number of criminal offences and who was also criticised by name by the district auditor. Mr Jones was suspended in August.
'Anglesey CC are not making any 'golden handshakes' or any other additional payment to Mr Gibson,' he said.