"We are taking this matter seriously, and together with our technical advisers we will be briefing our staff today," Ms Grant said.
Air tests have been carried out regularly over the past few years in the Clinical Services Building and all have come back negative.
"Safety remains the DHB's number one priority and staff briefings are currently taking place across the organisation," Southern DHB chief executive Carole Heatly said.
"We are taking expert advice on continued access to potentially affected parts of the hospital," she said.
"We are following best practice which might cause minor interruptions to staff and patients."
No essential services are affected.
Labour's acting health spokesman David Clark claimed that positive swab results have come from radiology on the first floor, a third floor laboratory, clinics and changing rooms on the fourth floor, and corridors on the fifth floor.
"That is horrifying," he said.
"Hospital staff should not be working in a building that makes them sick and risks shortening their lives, nor should patients and other members of the public be exposed to it."
He criticised Health Minister Jonathan Coleman for "dragging his heels" over the hospital's rebuild work.
"The asbestos threat makes the situation more urgent than ever," Mr Clark said.
"The Minister of Health needs to tell the public when a final business case for the hospital rebuild will be signed off by Cabinet, and he needs to commit to construction beginning by 2017 at the latest."
