Plunket northern region operations manager Linda Biddle said the health and safety of staff and the families Plunket worked with was of "paramount importance".
The change had not disrupted Plunket's services, she said.
The two parties had negotiated about a long-term solution for several months, with the council recently confirming it would remove the carpets in the worst affected room, sand and seal the timber floor and repaint the room.
It has also given Plunket approval to install an under floor ventilation system which the organisation will fund.
Plunket has asked the council to test the air quality tests inside the building after the work is done to check it has helped improve the environment.
Biddle said Plunket was working together with the council in a constructive way and is pleased to be able to continue providing services to the local community.
Kim O'Neill, Auckland Council's head of stakeholder and land advisory, said Plunket was a "valued tenant" and council wanted to "work constructively and collaboratively with them" to find a suitable solution to this issue.
"We regret this has taken longer than anticipated," O'Neill said.
Testing on the mould last September found it was not considered toxic, but could be allergenic.
A group room in the Plunket base was repainted, and all air vents and toilet exhaust grills were cleaned.
"Unfortunately, two months later, the mould reappeared," O'Neill said.
Current work included investigating any potential "inherent defects" in the building due to its age. It is more than 60 years old.
"The building itself has an insulated roof void, but no underfloor insulation," O'Neill said.
"There is adequate subfloor ventilation and there are adequate windows and doors. Plunket occupy an office adjacent to the room affected by the mould, and there is no significant mould in the area they do occupy."
All carpeting in the affected room is to be removed, and the original timber floor is to be polished and sealed to prevent mould from growing back.
"The council is currently working on options for new permanent premises for Plunket and will discuss potential options with Plunket Group," O'Neill said.