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The toxic material was unearthed last month by contractors working to upgrade playing surfaces and irrigation at the ground, council parks and recreation acting group manager Jendi Paterson said.
About 500 tonnes of soil had since been removed by specialists, and two sets of tests had confirmed the area was now safe, she said.
The cost of removing the asbestos would come from existing council budgets, and the wider upgrade was still on schedule to be completed in March, she said.
Some fences would remain up temporarily to protect newly-sown grass, she said.