But it may turn to external grant funding to pay for a giant glass case which is planned to surround the frontage of the former Doncaster Grammar School for Girls in the designs
Deputy mayor Glyn Jones confirmed the scheme had been approved after behind closed doors discussions on finance. It is possible the plan could need to find money from grants to to fund some of the costs.
Under the plans, the current library on Waterdale will be closed, along with the current Doncaster archives in Balby, and the schools library in Barnby Dun. The current museum on Chequer Road would become a storage facility for the new building.
The new building would include part of the remaining frontage of the former Doncaster Grammar School for Girls - but it emerged the cost of plans to put a glass case around the landmark school building's frontage could be £850,000.
It remains part of the plan.
Coun Jones said the current library site was expected to become housing or shops.
He said keeping the element of the former girls school was preserving Doncaster's heritage and creating a building that could possibly bring local and national awards.
"We will be trying to get grants from other bodies," he added.
He said: "Cabinet does like the glass encased design and we 'd like the cost to come from within the £14 million, but if needed extra external grants should meet any shortfall.
"I look forward to seeing this iconic building in Doncaster."
Other members of the cabinet also supported the plans for the new building, and cabinet member for culture, Coun Nigel Ball, said the new building was a step change to realise the council's ambitions.
He said: "It is a once in a lifetime chance to transform buildings from the 60s into something for the 21st century."