The work is expected to take about 16 months and be completed by September next year.
Ratepayers have already paid more than $128.5 million to buy and fit out the 25-year-old building, described as robust and structurally sound with good bones when it was bought in 2012.
Council general manager finance and property Kevin Ramsay said the the first due diligence review by Mott MacDonald in 2012, costing $90,000, was a largely visual inspection of stonework at the podium level and some imaging of the fixings behind the stone.
The review identified some cladding concerns for which $4.2 million was set aside, Mr Ramsay said.
The cost of undertaking a full invasive due diligence report for the whole building, Mr Ramsay said, would have been prohibitive.
Subsequent work in 2013 and 2014 during refurbishment work identified issues around the stone cladding and how it is fixed to the building.
A report by the engineering firm GHD in November 2015 found there was no definitive evidence of imminent collapse, but clearly cause for concern and a high potential risk of a stone panel falling from height.
The council is taking all reasonable steps to reduce or eliminate risk,including fencing and other measures around the facade.
Mr Ramsay said the council had already put aside $4.27 million to hang scaffolding at the top levels of the building and a concrete and steel reinforced working platform above the canopy as safety measures in the event of a granite slab falling.
There was nothing wrong with the structural stability of the building, or any concerns around weather-tightness, he said.
The decision to buy the ASB Tower was taken behind closed doors in December 2011 by the council's strategy and finance committee. It was hotly debated and passed by eight votes to six.
The eight councillors who supported the purchase were Alf Filipaina, Michael Goudie, Ann Hartley, Penny Hulse, Richard Northey, Sharon Stewart, Penny Webster, George Wood. Those opposed were Cameron Brewer, Cathy Casey, Sandra Coney, Mike Lee, Calum Penrose, Dick Quax.
Council officers recommended the "co-location of CBD staff as current lease agreements expire to reduce capital and operating costs over the longer term, improve financial and operational efficiencies, lift productivity and bring together critical business units".
Councillors were told the cumulative savings over 10 years wold be $27.5 million.